Integral inequalities of Hardy and Friedrichs types with applications ...
DOCTORA L T H E S I S
Integral Inequalities of Hardy and
Friedrichs Types with Applications
to Homogenization Theory
Yulia Koroleva
Integral Inequalities of Hardy and Friedrichs
Types with Applications to
Homogenization Theory
Yulia Koroleva
Department of Mathematics
Luleå University of Technology
SE971 87 Luleå, Sweden
korolevajula@mail.ru
16th June 2010
Key words: Inequalities, Homogenization Theory, Partial Differential
Equations, Functional Analysis, Spectral Theory, Friedrichstype Inequalities,
Hardytype Inequalities
Printed by Universitetstryckeriet, Luleå 2010
ISSN: 14021544
ISBN 9789174391046
Luleå 2010
www.ltu.se
Abstract
This PhD thesis deals with some new integral inequalities of Hardy and
Friedrichs types in domains with microinhomogeneous structure in a neighborhood
of the boundary. The thesis consists of six papers (Paper A – Paper
F) and an introduction, which put these papers into a more general frame
and which also serves as an overview of this interesting field of mathematics.
In papers A – D we derive and discuss some new Friedrichstype inequalities
for functions which belong to the Sobolev space H 1 in domains with
microinhomogeneous structure and which vanish on a part of the boundary
or on boundaries of small sets (cavities). The classical Friedrichs inequality
◦
H 1 with a constant depending only on the
holds for functions from the space
measure of the domain. It is well known that if the function has not zero
trace on the whole boundary, but only on a subset of the boundary of a positive
measure, then the Friedrichs inequality is still valid. Moreover, in such a
case the constant increases when the measure (the harmonic capacity) of the
set where the function vanishes, tends to zero. In particular, in this thesis we
study the corresponding behavior of the constant in our new Friedrichstype
inequalities. In papers E–F some corresponding Hardytype inequalities are
proved and discussed. More precisely:
In paper A we prove a Friedrichstype inequality for functions, having
zero trace on the small pieces of the boundary of a two  dimensional domain,
which are periodically alternating. The total measure of the set, where the
function vanishes, tends to zero. It turns out that for this case the capacity
is positive and hence the constant in the Friedrichs inequality is bounded.
Moreover, we describe the precise asymptotics of the constant in the derived
Friedrichs inequality as the small parameter characterizing the microinhomogeneous
structure of the boundary, tends to zero.
Paper B is devoted to the asymptotic analysis of functions depending on
the small parameter which characterizes the microinhomogeneous structure
of the domain where the functions are defined. We consider a boundaryvalue
problem in a twodimensional domain perforated nonperiodically along the
boundary in the case when the diameter of circles and the distance between
iii
iv
them have the same order. In particular, we prove that the Dirichlet problem
is the limit for the original problem. Moreover, we use numerical simulations
to illustrate the results. We also derive the Friedrichs inequality for functions
vanishing on the boundary of the cavities and prove that the constant in the
obtained inequality is close to the constant in the inequality for functions
◦
H 1 .
from
In paper C we consider a boundaryvalue problem in a threedimensional
domain, which is periodically perforated along the boundary in the case when
the diameter of the holes and the distance between them have the same order.
We suppose that the Dirichlet boundary condition is set on the boundary of
the cavities. In particular, we derive the limit (homogenized) problem for
the original problem. Moreover, we establish strong convergence in H 1 for
the solutions of the considered problem to the corresponding solutions of
the limit problem. Moreover, we prove that the eigenelements of the original
spectral problem converge to the eigenelements of the limit spectral problem.
We apply these results to obtain that the constant in the derived Friedrichs
inequality tends to the constant of the classical inequality for functions from
◦
H 1 , when the small parameter describing the size of perforation tends to
zero.
Paper D deals with the construction of the asymptotic expansions for
the first eigenvalue of the boundaryvalue problem in a perforated domain.
This asymptotics gives an asymptotic expansion for the best constant in a
corresponding Friedrichs inequality.
In paper E we derive and discuss a new twodimensional weighted Hardytype
inequality in a rectangle for the class of functions from the Sobolev space
H 1 vanishing on small alternating pieces of the boundary. The dependence
of the best constant in the derived inequality on a small parameter describing
the size of microinhomogenity, is established.
Paper F deals with a threedimensional weighted Hardytype inequality
in the case when the domain Ω is bounded and has nontrivial microstructure.
It is assumed that the small holes are distributed periodically along the
boundary. We derive a weighted Hardytype inequality for the class of functions
from the Sobolev space H 1 having zero trace on the small holes under
the assumption that a weight function decreases to zero in a neighborhood
of the microinhomogenity on the boundary.
Preface
This PhD thesis consists of six papers (Paper A – Paper F) and an introduction,
which put these papers into a more general frame and which also serves
as overview of these mathematical topics.
A. G. A. Chechkin, Yu. O. Koroleva and L. E. Persson, On the precise
asymptotics of the constant in the Friedrich‘s inequality for functions, vanishing
on the part of the boundary with microinhomogeneous structure, J.
Inequal. Appl., 2007, Article ID 34138, 13 pages, 2007.
B. G. A. Chechkin, Yu. O. Koroleva, A. Meidell and L. E. Persson, On the
Friedrichs inequality in a domain perforated nonperiodically along the boundary.
Homogenization procedure. Asymptotics in parabolic problems, Russ.
J. Math. Phys., 16 (2009), No.1, 1–16.
C. Yu. O. Koroleva, On the Friedrichs inequality in a cube perforated
periodically along the part of the boundary. Homogenization procedure,
Research Report 2, Department of Mathematics, Luleå University of Technology,
(25 pages), 2009.
D. G. A. Chechkin, R. R. Gadyl’shin and Yu. O. Koroleva, On the eigenvalue
of the Laplace operator in a domain perforated along the boundary,
Dokl. Math., 81 (2010), No.3, 1–5.
E. Yu. O. Koroleva, On the Weighted Hardy Type Inequality in a Fixed
Domain for Functions Vanishing on the Part of the Boundary, Research Report
1, Department of Mathematics, Luleå University of Technology, (12
pages), 2010.
F. Yu. O. Koroleva, On the Weighted Hardy Type Inequality in a Domain
with Perforation Along the Boundary, Research Report 4, Department of
Mathematics, Luleå University of Technology, (21 pages), 2010.
v
vi
Both the introduction and paper C are strongly influenced also by the following
paper:
G. Yu. O. Koroleva, On the Friedrichs inequality in threedimensional
domain perforated aperiodically along a part of the boundary, Uspekhi Math.
Nauk, 65 (2010), No.2, 181–182. [in Russian]
Acknowledgement
It is a pleasure for me to thank my main supervisors Professor LarsErik
Persson and Professor Gregory A. Chechkin for their attention to my work,
their valuable remarks and suggestions and for their constant support and
help.
Moreover, I thank my cosupervisor Professor Annette Meidell for helping
and supporting me in various ways e.g. concerning numerical and application
aspects on my work. I am deeply grateful to my cosupervisor Professor Peter
Wall for his carefully reading of my papers.
I thank my Russian colleagues from Moscow Lomonosov State University
for their interest to my research and for many fruitful discussions. I am also
grateful to PhD Student John Fabricius for helping me with several practical
things. Moreover, I appreciate very much the warm and friendly atmosphere
at the department of mathematics at Luleå University of Technology, which
helps to do research more effectively.
I also want to pronounce that the agreement about scientific collaboration
and PhD education between Moscow Lomonosov State University and Luleå
University of Technology has been very important. In particular, I thank
both of these Universities for financial support.
Finally, I thank my family for all support and just being there.
vii
Introduction
Integral inequalities of Friedrichs and Hardy types are very important for
different applications. In particular, they are often used for deriving some
estimates for operator norms, for proving some embedding theorems, for
solving various problems concerning PDE, Homogenization Theory, Spectral
Theory etc.
In this PhD thesis we prove and discuss some new integral inequalities of
Hardy and Friedrichs types in domains with microinhomogeneous structure
in a neighborhood of the boundary. In particular, the questions concerning
the optimal constants in the derived inequalities is considered. Moreover,
these inequalities are used to derive some new results in homogenization
theory.
The following estimate is known (see e.g. [76, Ch.III, §5]) as the Friedrichs
inequality for functions from the Sobolev space H ◦ 1 (Ω) :
∫
∫
u 2 dx ≤ K 0 ∇u 2 dx, (0.1)
Ω
where the constant K 0 depends only on the domain Ω. It should be noted that
the Friedrichs inequality first was obtained by K. Friedrichs for a bounded
domain Ω in the following form:
⎛
⎞
∫
∫
∫
u 2 dx ≤ K ⎝ ∇u 2 dx + u 2 ds⎠ ,
Ω
Ω
see [42]. Later, in the monograph [37] the sufficient conditions were given for
the validity of the following Poincaré inequality:
⎛ ⎞2
∫
∫
u 2 dx ≤ K ∇u 2 dx + 1 ∫
⎝ udx⎠
,
m Ω
Ω
Ω
where the constant m Ω depends on the domain Ω. Many mathematicians
have been interested in this inequality for various reasons. Therefore a lot of
Ω
∂Ω
Ω
ix
x
different results on the Friedrichstype and Poincarétype inequalities have
been obtained. In this introduction we will present some of the most important
of these results. In [97] a direct proof of the Friedrichstype inequalities
and the Poincarétype inequalities in some subspaces of Wp 1 (Ω) are given,
where Ω ⊂ R n is a bounded and connected open domain, n =2, 3. The
Poincaré and Friedrichs inequalities for any v ∈ Wω 1,p(Ω)
= {v ∈ W p 1(Ω) :
v ω =0}, 1 n.
We also mention the paper [77], where some properties of the best constant
in the Friedrichs inequality were studied. Here the following variational
Friedrichs inequality was considered:
min ‖∇u‖ L p(Ω)
‖u‖ Lq(Ω)
= λ pq (Ω) > 0inΩ, (0.2)
where the minimum is taken over all functions u ∈ Wp 1 (Ω) vanishing on
the whole boundary ∂Ω or on some part of it. Here it was assumed that
1 pthere exists ̂ε = ̂ε(q, p, n)
such that for ε
In the proofs a technique from the theory of the isoperimetrical inequalities
was used. In particular, the upper and the lower estimates for the best
constant was obtained. It was shown that the best constants in such types
of inequalities are inversely to a harmonic capacity (for a definition see, e.g.
[74, Ch.I, §2]) of the set where the function vanishes. The main result is
that the Friedrichs inequality holds if and only if the harmonic capacity is
positive.
In fact the calculation of this capacity is a nontrivial problem in the case
when the set has nontrivial microstructure. In this thesis we give direct
proofs of some Friedrichstype inequalities for functions from H 1 , vanishing
on the boundary of the cavities or on small pieces of a fixed boundary, and
show that the constants in these inequalities are close to the constants in
some ”limit” classical Friedrichstype inequalities.
In [92], [93] and [94] some discrete analogues of the Friedrichstype and
the Poincarétype inequalities were proved and discussed. In particular, in
[94] direct proofs of some discrete Poincaré and Friedrichs inequalities for
a class of nonconforming approximations of the Sobolev space H 1 (Ω) were
given:
∫
∑
∫
g 2 (x) dx ≤ C F ∇g(x) 2 dx, ∀g ∈ W 0 (J k ), (0.3)
K∈J k
∫
Ω
Ω
g 2 (x) dx ≤ C P
∑
∫
K∈J k
K
K
⎡
∫
∇g(x) 2 dx + ˜C p
⎣
Ω
⎤
2
g(x) dx⎦
, ∀g ∈ W (J k ).
(0.4)
Here {J k } k denotes a shaperegular simplicial triangulation of Ω. It consists
of triangles in a twodimensional space and of a tetrahedra in a threedimensional
space. The space W (J k ) consists of the functions, which locally
belong to H 1 (K) oneachK ∈J k , while the space W 0 (J k ) consists of all
functions from W (J k ), which have zero traces on the outer sides of K. These
spaces are nonconform approximations of the corresponding Sobolev spaces,
that is W 0 (J k ) H ◦ 1 (Ω) and W (J k ) H 1 (Ω). Moreover, in [94] also the best
constants in these inequalities were pointed out and the discrete Friedrichs
inequality was extended with a domain which is bounded only in one of the
directions. In twodimensional or threedimensional spaces it was proved
that the constant C F depends only on the square of the infimum over the
thickness of Ω in one direction. In addition, it was derived that C P depends
only on Ω both for convex and nonconvex domains Ω. One key idea in the
paper [94] was to extend the proof of the discrete Poincaré and Friedrichs
inequalities for the case with piecewise constant functions used in the finite
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xii
volume method to this more general case. It should be noted that the results
obtained in [94] can be applied in the analysis of nonconforming numerical
methods (e.g. for the discontinuous Galerkin method). Earlier, the validity
of (0.3) was established in [92] for W 0 (J k ) consisting of piecewise linear
functions. In [93] this result was generalized to spaces consisting of higher
order polynomial functions. Later, in [15] the inequalities (0.3) and (0.4)
were derived for piecewise H 2 functions in some twodimensional domains.
In paper A (see also [29]) we construct the asymptotics of the constant
in the Friedrichs  type inequality for functions from the Sobolev Space H 1 ,
which vanish on small pieces of the boundary. For this purpose some auxiliary
spectral problems corresponding to boundaryvalue problems with rapidly
alternating types of boundary conditions were studied and applied.
Boundaryvalue problems with rapidly alternating types of boundary
conditions were carefully studied by A.Damlamian, D.Q.Li, G.A.Chechkin,
M.E.Pérez, M.Lobo and O.A.Oleinik (see e.g. [16], [17], [18], [20], [21], [30],
[31], [32], [34], [38], [69] and [70]). Such problems can be described as follows:
the boundary of the domain is divided in two parts, some type of
boundary condition is set on one part of the boundary while the other part
is subjected to another boundary condition. Moreover, it is assumed that
one part of the boundary depends on a small parameter and consists of a
great number of nonintersecting components, which converge to zero (in the
sense of measure) when the small parameter tends to zero. The homogenization
procedures of such problems are now well studied (see e.g. [5], [16],
[20], [24], [25], [38], [39], [69] and [70]). The main result obtained in these
works can briefly be described as follows: the solutions of boundaryvalue
problems with rapidly alternating boundary conditions converge to the corresponding
solutions of boundary value problems with effective boundary
conditions. The type of limit boundary value problems depends on the ratio
between the measures of the parts of the boundary with different boundary
conditions in the original problems. The papers [20], [24], [27] and [30] deal
with the alternation of the Dirichlet boundary condition and the Neumann
or the Fourier (Robin) type of boundary conditions. Some estimates of the
rate of the convergence were obtained under the assumption that each small
component of the boundary shrinks into a point. In [20] the author considers
boundaryvalue problems with different types of boundary conditions, stated
on small alternating pieces of the boundary. The behavior of the solutions
of some problems when the small parameter tends to zero was studied and
described. In this paper the small parameter describes the period of changes
of types of the boundary conditions. In particular, the author obtains estimates
of the difference between the solutions of the original problems and the
solution of the corresponding limit problem. Moreover, the author studied
xiii
spectral properties of these problems. It should be noted that in this paper
for the first time it was presented a complete classification of the homogenized
problems, depending on the ratio of the small parameters characterizing the
frequency of the periodical change of the boundary conditions. In [24] some
boundaryvalue problems in multidimensional domains with rapidly alternating
boundary conditions were studied. In particular, it was proved that the
homogenized problems depend on the asymptotics of the first eigenvalue of
the corresponding spectral problem considered in the cell of periodicity. This
asymptotics was constructed by the authors and applied to estimate the rate
of the convergence of the solution of the original problems to the corresponding
solution of the homogenized problem when the small parameter tends to
zero.
In [25] some boundaryvalue problems with rapidly alternating types of
boundary conditions were considered for the Laplacian in a threedimensional
domain. It was assumed that the boundary of the domain consists of two
parts. One of them has a purely periodic microstructure. It could be rapidly
alternating spots or periodically distributed holes (for the case of unbounded
domain with perforated wall inside). In the first case we have a bounded
domain with micro inhomogeneous structure of the boundary, while in the
second one we have two domains connected through the holes. In the second
case the authors consider problems in both bounded subdomains or in
one bounded and one unbounded subdomain. In particular, the authors
presented a complete classification of the homogenized problems concerning
their dependence on the ratio of the small parameters characterizing the frequency
of the periodical change of the boundary conditions. Moreover, the
corresponding spectral problems were considered and both homogenization
and respective convergence theorems were proved.
Next we mention that the asymptotics of the solutions of some problems
with rapidly oscillating boundary conditions were constructed in [7], [8], [9],
[10], [11], [14], [21], [43], [44] and [45]. Furthermore, some twodimensional
boundary value problems were considered in [8], [9], [14], [43], [44] and [45].
In [21] the asymptotic expansion was constructed for the solution of the
Poisson equation in a multidimensional layer in the case when the boundary
conditions alternate periodically on sets shrinking into a point. Moreover,
complete expansions for the eigenelements of the Laplace operator in
a cylinder with frequently alternating Dirichlet and Neumann conditions on
the surface of the cylinder were constructed in [7] and [10]. In [10] a special
problem with Dirichlet condition on the lateral surface was considered as
homogenized. It was assumed that the pieces of the boundary with Dirichlet
condition have the same order of small size as pieces with Neumann conditions.
In [7] the case corresponding to limit problems with Neumann or
xiv
Fourier (Robin) boundary conditions was studied. The author proved both
in [10] and [7] that the original problem has only eigenvalues of multiplicity
one or two. Moreover, in [7] the author constructed explicitly the leading
terms of the asymptotic expansions of the eigenvalues and the eigenfunctions
in the case when the boundary value problem has Neumann or Fourier
(Robin) boundary condition on the lateral surface of a cylinder having an
arbitrary section. In addition, it was shown that these eigenvalues converge
to the corresponding limit eigenvalues of multiplicity one. In [13] a singularly
perturbed boundary value spectral problem for the Laplacian in a cylinder
with frequently alternating boundary conditions on the lateral surface was
studied. The lateral surface was divided into a great number of thin strips
on which the Dirichlet and the Neumann conditions are alternating. The
case when the homogenized problem has Dirichlet boundary condition on
the lateral surface was considered. The leading terms of the asymptotic expansions
of the eigenelements in the case of strips with slowly varying width
were constructed. Moreover, for the case with strips with frequently varying
width, some estimates for the rate of convergence were derived. The results
obtained in [13] were also discussed in [11].
In paper A (see also [29]) we use the convergence of the eigenvalues of
some boundaryvalue problems with rapidly alternating types of boundary
conditions to obtain an independently interesting fact concerning the constant
in the Friedrichs inequality.
The papers B and C are devoted to investigations of some new boundary
value problems with perforation along the boundary.
Boundaryvalue problems in perforated domains were studied by many
authors (see e.g. [2][4], [23][40], [55], [56], [67][75] and [81][89]). In these
papers and monographs the authors studied different kinds of perforation for
linear as well as nonlinear differential operators. In the present introduction
we briefly describe the obtained results.
Boundaryvalue problems in perforated domains in the case when the
cavities are distributed periodically with a small period and provided that
the size of the cavities has the same order as the period of perforation, was
studied in works of O.A. Oleinik and her students (see e.g. [54], [80] and
[81]). The case when the size of the cavities has much less order then the
period of perforation was considered in [36], [55] and [56].
The paper [3] (see also the short note [4]) was dedicated to analysis of
boundaryvalue problems in perforated domains in the case when the diameter
of the holes is much less then the distance between them. A homogenization
theorem was proved for a boundaryvalue problem with Neumann
boundary conditions on the external boundary and with Dirichlet boundary
conditions on the boundary of cavities. Moreover, the author constructed the
first term of the asymptotic expansion of the original problem and obtained
estimates of the difference between the original solution and the first term
of the asymptotic expansion. The author studied also spectral properties of
such boundary value problems and obtained some estimates of the difference
between the eigenelements of the original problems and the eigenelements of
the corresponding homogenized problems.
In [75] the Dirichlet problem was considered in a domain with aperiodically
distributed cavities. In particular, the weak convergence in L 2 of the
solutions in terms of convergence of a harmonic capacity of the cavities was
proved.
Boundaryvalue problems in domains with perforation along a surface
were studied in [40], [71], [72, Ch. I, §3] and [85].
In [72, Ch. I, §3] the author studied a boundaryvalue problem in a
domain perforated along a closed curve. In particular, the uniform convergence
of the solutions of the original problems in compact subdomains,
involving the curve, to the solutions of a limit problem was proved. In [85] a
boundaryvalue problem was stated in a domain divided by a perforated wall.
Moreover, weak convergence of the solutions of the homogenized problems to
the solution of a limit problem was proved. Moreover, the estimates of the
rate of the convergence for some special cases were obtained. In [40] the author
considered the thick Neumann sieve problem, which is an extension of a
problem raised by V.O. Marchenko and E.Ya. Khruslov (see [72]) and, more
specifically, by E. SanchesPalencia (see [85]): the plane x n =0, perforated
by εperiodically distributed small holes with diameters r(ε) < ε , separates
2
an ndimensional bounded domain G into two subdomains G + and G − . The
behavior of the solutions u ε of boundaryvalue problems in G was studied
when ε → +0. In [40] an analogous problem was considered for a “thick”
sieve enlarged to thickness h(ε) with holes being cylinders of height h(ε) (see
Figures 0.1 and 0.2).
xv
Figure 0.1: The sieve in the case n =2.
The function u ε is the solution of the equation −Δu + u = f subject
to the Neumann boundary condition on the surface of the sieve and, e.g.,
the Dirichlet condition on the boundary of G without sieve. The solution
u ε converges, in some sense, to u + in G + and to u − in G − when ε, h(ε) and
r(ε) tend to 0. It was proved in [40] that these functions are the solutions of
the equation above and satisfying the Dirichlet condition on the boundary
xvi
Figure 0.2: The sieve in the case n =3.
of G without sieve, but along the plane x n =0, jumps arise according to the
following:
∂u +
∂n + 1 2 μ(u+ − u−) =0,
where 0 ≤ μ ≤ +∞ is some constant and n is the outer normal with respect
to G + and G − . Moreover, the dependence of the constant μ on the functions
r(ε)andh(ε)whenε → 0andfordimensionn ≥ 3 was derived and discussed.
In particular, this means that μ = 0 if the holes have “too small diameters”
or are “too long”; μ>0 if the holes are “big enough” but “not too long” (in
the later case μ = ∞). These cases are exactly separated by some critical
functions r ∗ (ε),h ∗ (ε) and the values of μ are given by a functional looking
very similar as the definition of a capacity. However, for the dimension
n = 2 some difficulties arise in the analogous problems so that some questions
remain open.
The paper [71] is devoted to the asymptotic behavior of the solutions of
a boundaryvalue problem in a domain perforated along a manifold with
different types of conditions on the boundary of cavities. Here the authors
consider a bounded domain Ω in R n with sufficiently smooth boundary
∂Ω, and a set γ, which is a collection of some sets Ω. Let the points
P j (j = 1,...,N(ε), N(ε) ≤ d 0 ε 1−n ,d 0 = const) belongtoγ, where ε as
usual denotes a small positive parameter. Denote by G j (a j ε )asetbelonging
to Ω, with smooth boundary ∂G j (a j ε), P j ∈ G j (a j ε), the diameter of G j (a j ε)
equals to a j ε ,aj ε ≤ C 0ε, C 0 = const > 0,G j (a j ε ) ∩ Gi (a i ε )=∅ for i ≠ j. The
authors consider all possible cases of behavior of a j ε when ε → 0.
Let us describe the results of [71] in more detail. The authors used the
following notations:
G ε =
N(ε)
⋃
j=1
N(ε)
G j (a j ε ), Ω ε =Ω\ G ε ,S ε ‘ = ⋃
j=1
∂G j (a j ε ),
xvii
S ε =
N(ε)
⋃
j=1
∂G j (a j ε) ∩ Ω, Γ ε = ∂Ω ε \ S ε .
The following boundary values problems were considered:
{
−Δu ε = f in Ω ε ,
∂u ε
∂ν =0onS ε, u ε =0onΓ ε ,
(0.5)
or
{
−Δu ε = f in Ω ε ,
∂u ε
+ βu ∂ν ε =0onS ε , u ε =0onΓ ε , β(x) ≥ β 0 = const > 0,
or {
−Δu ε = f in Ω ε ,
u ε =0on∂Ω ε ,
where ν denotes the unit outer normal to S ε .
Let v 0 be a solution of the following problem:
{
−Δv 0 = f in Ω,
v 0 =0on∂Ω,
(0.6)
(0.7)
(0.8)
where f is a smooth function from Ω.
The following estimates were obtained in the case when u ε is the weak
solution of the problem (0.5), and v 0 is the solution of (0.8):
Theorem 0.1. (1). Assume that S ε ‘ ∩ ∂Ω =∅. Then
‖u ε − v 0 ‖ 2 H 1 (Ω ≤ K ε) 1(max a j ε) n ε 1−n ,
j
and
λ k ε − λ k  2 ≤ C 1 (max a j ε) n ε 1−n .
j
(2). Assume that S ε ‘ ∩ ∂Ω ≠ ∅ and that the number of sets Gj (a j ε ),
nonintersecting with ∂Ω, is less then or equal to d 1 ε 2−n , and that
∂G j (a j ε ) ≤ d 2(a j ε )n−1 . Then
‖u ε − v 0 ‖ 2 H 1 (Ω ≤ K ε) 2 max(a j ε
j
)n−1 ε 2−n ,
and
λ k ε − λk  2 ≤ C 1 max(a j ε
j
)n−1 ε 2−n .
Here {λ k ε }, {λk },k =1, 2,... are the sequences of eigenvalues of the corresponding
spectral problems.
xviii
The convergence described in our next Theorem was proved in the case
when u ε is the weak solution of the problem (0.6) and v 0 is the solution of
the limit problem (0.8):
Theorem 0.2. Assume that ∂G j (a j ε) ≤ K(a j ε) n−1 , and that
η ε ≡ (max a j ε
j
)ε−1 → 0 when ε → 0. Then ‖u ε − v 0 ‖ 2 H 1 (Ω ε)
→ 0
when ε → 0, and
‖u ε − v 0 ‖ 2 H 1 (Ω ≤ K ε) 3ηε n−1 ,
and
) n−1
λ k ε − λ k  2 ≤ C 2
(max (a j ε)ε −1 .
j
Here {λ k ε}, {λ k },k =1, 2,... are the sequences of eigenvalues of the corresponding
spectral problems.
Furthermore, the authors considered the case when γ is a domain on
the hyperplane {x : x 1 = 0}, γ = Ω ∩{x : x 1 = 0}, Q = {x :
− 1 < x
2 1 < 1 ,j = 1,...,n}, 2 G‘ ε = ⋃ (a ε G 0 + εz). Here G 0 is a
domain with a smooth boundary, a ε ≤ Cε, C = const, G 0 ⊂ Q, and Z is
the set of vectors with integer components. In addition, it was assumed that
a ε ε −1 →C 0 = const > 0whenε → 0andC 0 G 0 ⊂ Q.
Let the domain Ω ε be represented in the following union:
z∈Z
Ω ε =Ω + ε ∪ γ+ ε ∪ Π∗ ε ∪ γ− ε ∪ Ω− ε , (0.9)
where
{
Π ε =Ω∩ x : x 1  < ε }
{
, Π ∗ ε
2
=Π ε \ G ‘ ε ,γ± ε =Ω∩ x : x 1 = ± ε }
,
2
{
Ω + ε =Ω∩ x : x 1 > ε }
{
, Ω − ε
2
=Ω∩ x : x 1 < − ε }
, and G ε = G ‘ ε
2
∩ Ω.
Besides this, S ε = ∂G ε ∩ Ω, Γ ε = ∂Ω ε \ S ε ,l ε = G ε ,l ε = G ε ∩ ∂Ω.
The authors studied the problem
{
−△v = f in Ω \ γ, v =0on∂Ω,
[v] γ
=0, [ ∂v
∂ν]∣
∣γ = μv γ
,
(0.10)
where μ = β 0 C n−1
0 ∂G 0 .
If u ε is a weak solution of the problem (0.6) in Ω ε , given by formula (0.9),
and v is a solution to the problem (0.10), then the following Theorem holds.
Theorem 0.3. Suppose that a ε ε −1 →C 0 = const > 0 when ε → 0, and that
l ε ≤d 3 ε. Then
and
‖u ε − v‖ H 1 (Ω ε) ≤ K 4
(√ ε + aε ε −1 −C 0  ) ,
λ k ε − λk  2 ≤ C 3
(
ε + aε ε −1 −C 0  2) .
Here {λ k ε}, {λ k },k =1, 2,... are sequences of eigenvalues of the corresponding
spectral problems.
The Dirichlet problem for the Poisson equation was also considered. Let
u ε be the weak solution of (0.7) and let v be the solution of the problem
(0.8). Then the following Theorem is valid.
Theorem 0.4. Suppose that
⎧
⎨lim
max (a j ε
ε→0 j
)n−2 ε 1−n =0, when n ≥ 3,
⎩lim
ε −1 ln a j ε −1 =0, when n =2.
max
ε→0 j
xix
If n ≥ 3, then
‖u ε − v 0 ‖ 2 H 1 (Ω ≤ K ε) 5(max a j ε) n−2 ε 1−n ,
j
and if n =2, then
λ k ε − λ k  2 ≤ C 4 (max a j ε) n−2 ε 1−n ,
j
‖u ε − v 0 ‖ 2 H 1 (Ω ε) ≤ K 6 max
j
ε ∣ −1 ∣ln a j ∣ −1 ε ,
λ k ε − λ k  2 ≤ C 5 max ε ∣ ∣ −1 ln a
j
ε −1
.
j
Here {λ k ε }, {λk },k =1, 2,... are sequences of eigenvalues of the corresponding
spectral problems.
Moreover, the problem (0.7) was considered in the domain Ω ε , determined
by formula (0.9) under the assumption G 0 = {x : x 0, when n ≥ 3,
⎩lim
ε→0 ε−1 ln a ε  −1 = C 2 = const > 0, when n =2.
xx
In this case the limit problem is the following one:
⎧
⎨−△v = f
[
in Ω
]∣ − ∪ Ω + ,
∂v ∣∣γ
⎩[v] γ
=0,
∂x 1
= μ 1 v γ
,v=0on∂Ω,
(0.11)
where Ω − =Ω∩{x : x 1 < 0}, Ω + =Ω∩{x : x 1 > 0},μ 1 =(n−2)a n−2 ω(n)C 1 ,
if n ≥ 3, μ 1 =2πC 2 , if n =2, where ω(n) is the square of the unit sphere in
R n .
The following estimates are obtained for this case.
Theorem 0.5. Suppose that a ε ε −1 →C 0 = const > 0 when ε → 0, and that
l ε ≤d 3 ε. Then
‖u ε − v‖ L2 (Ω ε) ≤ K 7
(√ ε + C1 − a n−2
ε ε 1−n  ) ,
and
when n ≥ 3 and
λ k ε − λ k  2 ≤ C 6
(
ε + C1 − a n−2
ε ε 1−n  2)
(√
‖u ε − v‖ L2 (Ω ε) ≤ K 8 ε + C2 − (ε ln a ε ) −1  ) ,
(
λ k ε − λ k  2 ≤ C 7 ε + C2 − (ε ln a ε  2 ) −1  )
when n =2. Here {λ k ε}, {λ k },k =1, 2,... are the sequences of eigenvalues of
the corresponding spectral problems.
Furthermore, the Dirichlet problem (0.7) was considered in [71] in the
domain Ω ε defined by (0.9) under some additional assumptions:
{
a 2−n
ε ε n−1 → 0whenε → 0, if n ≥ 3,
ε ln a ε →0, when ε → 0, if n =2.
Here the authors assumed that a j ε = a ε. In this case the limit problem is as
follows: {
−△v − = f in Ω − ,v=0on∂Ω − ,
−△v + = f in Ω + ,v=0on∂Ω + .
In this case the following estimates of the rate of the convergence hold.
Theorem 0.6. It yields that
‖ũ ε − v ± ‖ 2 L 2 (Ω ε) ≤ K 9a 2−n
ε ε n−1 ,
xxi
and
when n ≥ 3 and
and
λ k ε − λk  2 ≤ C 8 a 2−n
ε ε n−1
‖ũ ε − v ± ‖ 2 L 2 (Ω ε) ≤ K 10 ε ln a ε  ,
λ k ε − λk  2 ≤ C 9 ε ln a ε 
when n =2. Here ũ ε denotes an extension of u ε in Ω, and {λ k ε }, {λk } are the
sequences of eigenvalues of the corresponding spectral problems.
Boundaryvalue problems in perforated domains for nonlinear differential
operators were considered e.g. in [67], [68], [86], [87], [88] and [89]. The
monograph [86] deals with boundaryvalue problems for nonlinear elliptic
equations of an arbitrary order. The first half of the book deals with the
solvability problem while the second one is devoted to the properties of the
weak solutions. Moreover, the author considers homogenization of a family
of quasilinear boundaryvalue problems both in domains with a finegrained
boundary and in domains with channels. In particular, it was proved that the
solutions of these boundaryvalue problems are close to a solution of a certain
limit quasilinear problem in a nonperforated domain. The Dirichlet problem
in perforated domains for the nonlinear equations was considered e.g. in
[86], [87] and [88]. Moreover, further investigations of quasilinear problems
in a sequence of domains with a finegrained boundary and a careful study
of the convergence of the eigenvalues of some nonlinear Dirichlet problems
in such domains can be found in the paper [89]. Here a domain as follows
⋃
was considered: Ω s =Ω\
F (s)
i
I(s)
i=1
F (s)
i , where Ω ⊂ R n is an arbitrary domain and
,i=1,...I(s) < ∞, s∈ N, is a finite number of disjoint closed domains,
which are contained in Ω. Here it is also assumed that the diameters of the
sets F (s)
i , the distance between two neighboring sets and the distance between
the set and the boundary of Ω tend to zero when s →∞. In this domain the
following nonlinear problem was studied:
⎧
⎪⎨
⎪⎩
n∑
j=1
d
dx j
f j (x, u s (x), ∇u s (x)) − f 0 (x, u s (x), ∇u s (x)) =
= λ s g 0 (x, u s (x)), x ∈ Ω s ,
u s (x) = 0, x ∈ ∂Ω s .
(0.12)
It was proved that the problem below is the limit of (0.12) under some con
xxii
ditions on functions g 0 ,f j ,j=0,...,n:
⎧ n∑
d
⎪⎨ dx j
f j (x, u(x), ∇u(x)) − f 0 (x, u(x), ∇u(x)) + c 0 (x, −u(x)) =
j=1
= λg 0 (x, u(x)), x ∈ Ω,
⎪⎩
u(x) = 0, x ∈ ∂Ω.
(0.13)
Moreover, the convergence of the eigenelements of the homogenized problem
to the eigenelements of the corresponding limit problem was established: let
λ s be the eigenvalue of (0.12) and λ be the eigenvalue of (0.13) and let u s (x)
and ũ(x) be the corresponding eigenfunctions. Then lim λ s = λ and the
s→∞
sequence {u s (x)} ∞ s=1 converges when s →∞to ũ(x) strongly in Wr 1 (Ω) for
any r
xxiii
boundary: {
Lξ = η
(η, x) =(ξ Γ ,x)
in S
x ∈ X + (Γ),
(0.14)
where the mappings η : x(S) → H, ξ Γ : X + (Γ) → H are linear and continuous
from one side, and the following deterministic problem with the random
boundary from the other side:
{
Lξ n =0 inG n ,
(0.15)
(ξ , x)=(ξ,x) x ∈ X + (∂G n ),
where G n ⊆ S, ξ is the solution of (0.14) and ξ n = ξ in S \ G n . The main
result is the following theorem.
Theorem 0.7. The weak convergence ξ n → u in H =(Ω,A,P) holds P ′ ,
a.s., if
lim
n→∞ nεd n = α, 0 ≤ α
xxiv
as a random variable μ j (w 1 ,...,w N ) on the product space M ×···×M (N
times). For β = 1 it was shown in [72] that μ j ({w}) converges a.e. to the
jth eigenvalue μ V j of the Schrödinger operator H = −Δ+4παV in M, where
V denotes the distribution density of each variable w k . The authors in [82]
refined this result and established a central limit theorem for the random
variables μ j ({w k })whenm →∞. It asserts that μ j ({w k })=μ V j + m β 2 −1 S j ,
with Gaussian variables S j of zero mean and the variance depending on the
jth eigenfunction of Δ M . The authors also presented an approximate construction
of the Green function G(x; y) ofΔ M\∪Bk , and precise estimates of
the difference between G and its approximation.
The asymptotic behavior of the solution in a domain randomly perforated
along the boundary was studied in [23] (see also the short note [33]). It was
proved that the solutions of the original problems with a random structure
converge weakly in the Sobolev space H 1 (Ω) to the solution of a nonrandom
(deterministic) problem with homogenized boundary conditions on the
boundary of the domain.
The paper B is devoted to asymptotic analysis of functions depending
on a small parameter describing the microinhomogeneous structure of the
domain, where the functions are defined. We consider a boundaryvalue
problem in a rectangle perforated nonperiodically along the boundary in the
case when the diameters of circles and the distance between them are of the
same order. The main result obtained in paper B is the convergence of the
solutions of the original problem to the solution of the corresponding limit
problem as well as the convergence of the eigenelements of the respective
spectral problems. Moreover, we apply this result to estimate the constant
in our new Friedrichstype inequality.
In paper C (see also [59] and [62]) we consider a problem in a threedimensional
cube, periodically perforated along a part of the boundary in the
case when the diameter of holes and the distance between them have the same
order. An analogous problem for a twodimensional domain was considered
in [48] (periodic case) and in [28] (aperiodic case). Here we suppose that
the Dirichlet boundary condition is set on the boundary of cavities. The
asymptotic behavior of the solutions of the original problem is studied in
detail. In particular, we derive the limit (homogenized) problem for the
original problem. Moreover, we establish strong convergence in H 1 of the
solutions of the considered problem to the corresponding solution of the limit
problem. Moreover, we prove that the eigenelements of the spectral problem
converge to the eigenelements of the spectral limit problem. Besides this, we
give the direct proof of the Friedrichs inequality for functions belonging to the
Sobolev space H 1 , vanishing on small cavities of the perforation. Moreover,
we prove that the difference between the constants in our new and classical
xxv
Friedrichs inequality tends to zero when the small parameter tends to zero.
We derive this fact by using the convergence of the eigenvalues of the original
and the homogenized problem considered in this paper.
Paper D (see also [26]) deals with deriving of the asymptotic expansions
for the first eigenvalue for a boundaryvalue problem in a perforated domain.
This asymptotics gives an asymptotical expansion for the best constant in a
corresponding Friedrichs inequality.
The Friedrichs inequality is a special case of the Hardy inequality. We
continue by giving a short review of Hardytype inequalities.
The classical Hardy inequality was first published (without proof) by G.
H. Hardy in 1920 in [50] and proved in 1925 in [52]. Also the first weighted
version of Hardy’s inequality is due to Hardy himself and reads as follows
(see [51]):
∫ ∞
( ) p ∫∞
p
x −r F (x) p dx ≤
x −r (xf(x)) p dx, (0.17)
r − 1
0
where f(x) ≥ 0, p>1, r≠1and
F (x) =
∫ x
0
f(t) dt as r>1, F(x) =
0
∫ ∞
x
f(t) dt as r1andu(x) =F (x) weobtain
the following so called Hardy inequality in differential form:
∫ ∞ ( )
u p
p ∫∞
x dx ≤ p
u′ p x a dx, 1
xxvi
which is very useful for many applications. This inequality obviously hold for
u(x) ∈ AC[0, ∞), u(0) = 0, where AC[0, ∞) is the class of absolutely continuous
on [0, ∞) functions. Equality in (0.18) holds if and only if u(x) = 0.
In this thesis we prove some new multidimensional Hardytype inequalities
and we here just mention a few classical results, which can serve as a
frame of these new inequalities.
It was J. Nečas who first generalized the inequality (0.18) to domains
in R n (see [78]). He proved that if Ω is a bounded domain with Lipschitz
boundary, 1
xxvii
In particular, for n = 1 the precise result reads as follows: The inequality
∫ ∞
∫ ∞
q(x)u 2 (x) dx ≤ c (u ′ ) 2 (x) dx, where u(0) = 0,
0
0
holds if and only if there exists a constant C 1 such that
sup x
x∈(0,∞)
∫ ∞
x
q(x) dx ≤ C 1 .
In papers E and F we derive and apply some new Hardytype inequalities
of the type ∫
∫
ρ α−2 (x)u(x) 2 dx ≤ C ρ α (x)∇u(x) 2 dx (0.20)
Ω
in the case when Ω is bounded and has a nontrivial microstructure. We
assume that the weight function ρ = ρ(x) is decreasing to zero when x
approaches the part of the boundary with the microinhomogenity.
In paper E the inequality (0.20) is proved under the assumption that the
functions u ∈ H 1 vanishes on small alternating pieces of a part of the boundary
of a fixed domain. It is assumed for the simplicity that Ω is a rectangle in
R 2 . But the same result is still valid in the case when Ω is an arbitrary domain
with a sufficiently smooth boundary with a microinhomogeneous structure.
In paper F (see also [61]) we derive the inequality (0.20) under the assumption
that Ω is a cube with a periodical perforation along a part of the
boundary. The Dirichlet condition holds on the boundaries of small bolls,
while the Neumann condition is assumed on the lateral surface of the cube.
It should be mentioned that results in this direction are completely new
in the theory of Hardytype inequalities. In particular, this gives us a possibility
to use ideas and techniques developed within homogenization theory to
obtain the estimates for the best constants in this type of Hardy inequalities.
Finally, we mention that some new open problems are presented in this
PhD thesis, which can be of interest in further research in this fascinating
field of mathematics.
Ω
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Paper A
On the Precise Asymptotics of the Constant
in the Friedrich’s Inequality for Functions
Vanishing on the Part of the Boundary with
Microinhomogeneous Structure 1
G.A. Chechkin, Yu.O. Koroleva and L.E. Persson
Recommended by Michel Chipot
Abstract
We construct the asymptotics of the sharp constant in the Friedrich
type inequality for functions, which vanish on the small part of the
boundary Γ ε 1 . It is assumed that Γε 1 consists of ( )
1 n−1
δ pieces with
diameter of order O(εδ). In addition, δ = δ(ε), and δ → 0asε → 0.
2000 Mathematics Subject Classification: 26D10, 26D15, 26B99
Key words: Inequalities, Friedrich’s inequality, Hardy Type inequalities,
asymptotics of sharp constants.
Introduction.
The domain Ω is an open bounded set from the space R n . The Sobolev
space H 1 (Ω) is defined as the completion of the set of functions from the
√ ∫
space C ∞ (Ω) by the norm (u 2 + ∇u 2 ) dx. The space H ◦ 1 (Ω) is the set
Ω
of functions from the space H 1 (Ω), with zero trace on ∂Ω.
Let ε = 1 , N ∈ N, be a small positive parameter. Consider the set
N
Γ ε ⊂ ∂Ω which depends on the parameter ε. The space H 1 (Ω, Γ ε )istheset
of functions from H 1 (Ω), vanishing on Γ ε .
1 The work of the first author was partially supported by RFBR (060100138). The
work of the first and the second authors was partially supported by the program “Leading
Scientific Schools” (HIII2538.2006.1). The work of the second author was partially
supported by RFBR.
1
The following estimate is known as the Friedrich’s inequality for functions
u ∈ H ◦ 1 (Ω) :
∫
∫
u 2 dx ≤ K 0 ∇u 2 dx, (0.1)
Ω
where the constant K 0 depends on the domain Ω only and does not depend
on the function u.
Inequality (0.1) is very important for several applications and it may
be regarded as a special case of multidimensional Hardy–type inequalities.
Such inequalities has attracted a lot of interest in particular during the last
years, see e.g. the books [12], [13] and [17] and the references given there.
We pronounce that not so much is known concerning the best constants in
multidimensional Hardy–type inequalities and the aim of this paper is to
study the asymptotic behavior of the constant in [5] for functions vanishing
on a part of the boundary with microinhomogeneous structure. In particular,
such result are useful in Homogenization Theory and in fact this was our
original interest in the subject.
The paper is organized as follows: In Section 1 we present and discuss
our main results. In Section 2 these results are proved via some auxiliary
results, which are of independent interest. In Section 3 we consider partial
cases, where it is possible to give the asymptotic expansion for the constant
with respect to ε.
Ω
1 The main results.
It is well known (see, for instance [15]) that the ( Friedrich’s ) inequality (0.1)
is valid for functions u ∈ H 1 1
(Ω, Γ ε )andK 0 = O
cap Γ ε
, wherewedenote
by cap F the capacity of F ⊂ R n in R n
{ ∫
}
cap F =inf ∇ϕ 2 dx : ϕ ∈ C0 ∞ (R n ),ϕ≥ 1onF .
R n
Remark 1. The Friederich’s inequality, when the functions vanishes on a
part of the boundary is sometimes called “Poincaré’s inequality”, but we prefer
to say “Friedrich’s” or “Friedrich’s type inequality” keeping the name
“Poincaré’s inequality” for the following (see, for instance [8]):
∫
( ∫
u 2 dx ≤
2 ∫
udx)
+
∇u 2 dx,
∀ u ∈ ◦
H 1 (Ω).
Ω
Ω
Ω
2
Further, it will be shown later on that K 0 is uniformly bounded under
special assumptions on Γ ε in the case when mes Γ ε → 0asε → 0.
Consider now the domain Ω ⊂ R 2 with smooth boundary of the length
1, such that
∂Ω =Γ ε 1 ∪ Γε 2 , Γε 1 = ⋃ i
(Γ ε 1i ), Γε 2 = ⋃ i
(Γ ε 2i ), Γε 1i ∩ Γε 2j = ∅,
( ) 1
mes Γ ε 1i = εδ(ε), mes (Γε 1i ∪ Γε 2i )=δ(ε), δ(ε) =o ln ε
as ε → 0,
where Γ ε 1i and Γε 2i
are alternating (see Figure 1).
Figure 1: Spatial domain
Here Γ ε =Γ ε 1 . Our first main result reads.
Theorem 1.1. Suppose n =2.Foru ∈ H 1 (Ω, Γ ε 1 ) the following Friedrich’s
inequality:
∫
∫
u 2 dx ≤ K ε ∇u 2 dx, K ε = K 0 + ϕ(ε),
Ω
Ω
holds true, where K 0 is a constant in the Friedrich’s inequality (0.1) for
functions u ∈ H ◦ 1 (Ω), andϕ(ε) ∼ (ln ε) − 1 1
2 +(δ(ε) ln ε)
2 as ε → 0.
For the case n ≥ 3 the geometrical constructions are similar. We assume,
that ∂Ω =S ∪ Γ, S∩ Γ=∅, Γ belongs to the hyperplane x n =0and
Γ=Γ ε 1 ∪ Γε 2 , Γε 1 ∩ Γε 2 = ∅. 3
Denote by ω a bounded domain in the hyperplane x n =0, which contains
the origin. Without loss of generality ω ∈ ✷,
✷ = {ˆx : − 1 2
2 Proofs of the main results and some auxiliary
results.
In Subsections 2.1 and 2.2 of this Section we discuss, present and prove some
auxiliary results, which are of independent interest but also crucial for the
proof of the main results in Subsection 2.3.
2.1 The relation between the constant in the Friedrich’s
inequality and the first eigenvalue of a boundary
value problem.
Let Ω be some bounded domain with smooth boundary ∂Ω, Γ ε ⊂ ∂Ω. Suppose
that the function u belongs to H 1 (Ω). Consider the following problem:
⎧
⎪⎨ Δu = −λ ε u in Ω
u =0 onΓ ε
(2.1)
⎪⎩ ∂u
=0 on∂Ω \ Γ ∂ν ε.
Definition 2.1. The function u ∈ H 1 (Ω, Γ ε ) is a solution of the Problem
(2.1), if the following integral identity:
∫
∫
∇u∇vdx= λ ε uv dx
Ω
Ω
is valid for all functions v ∈ H 1 (Ω, Γ ε ).
The operator of the Problem (2.1) is positive and selfadjoint (it follows
directly from the integral identity). According to the general theory (see, for
instance, [19]), all eigenvalues of the problem are real, positive and satisfy:
0 ≤ λ 1 ε ≤ λ 2 ε ≤ ..., λ k ε →∞as k →∞.
Here we assume that the eigenvalues λ k ε are repeated according to their multiplicities.
Denote by μ ε the following value :
∫
∇v 2 dx
Ω
μ ε = inf ∫
v∈H 1 (Ω,Γ ε)\{0} v 2 dx . (2.2)
We need the following Lemma (see the analogous Lemma in [5]):
5
Ω
Lemma 2.1. The number μ ε is the first eigenvalue λ 1 ε of the problem (2.1).
For the convenience of the reader we present the details of the proof.
Proof. It is sufficient to show that there exists such eigenfunction u 1 of the
Problem (2.1), corresponding to the first eigenvalue λ 1 ε , that it satisfies
∫
∇u 1  2 dx
Ω
μ ε = ∫
(u 1 ) 2 dx .
Let {v (k) } be a minimization sequence for (2.2), i.e.
Ω
v (k) ∈ H 1 (Ω, Γ ε ), ‖v (k) ‖ 2 L 2 (Ω)
∫
=1, and
∇v (k)  2 dx → μ ε , as k →∞.
Ω
It is obvious that the sequence {v (k) } is bounded in H 1 (Ω, Γ ε ). Hence, according
to the Rellich theorem, there exists a subsequence of {v (k) },converging
weekly in H 1 (Ω, Γ ε ) and strongly in L 2 (Ω). For this subsequence we keep the
same notation {v (k) }.Wehavethat
‖v (k) − v (l) ‖ 2 L 2 (Ω) k 0(η).
Using the following formula
v (k) + v (l) 2
∥ 2 ∥ = 1 ∥ ∥ v
(k) 2
L 2 (Ω)
2
+ 1 ∥ ∥ ∥ L 2 (Ω) v
(l) ∥∥∥ 2
2
− v (k) − v (l)
L 2 (Ω)
2 ∥
2
L 2 (Ω)
we obtain that ∥ ∥∥∥ v (k) + v (l) 2
2 ∥ > 1 − η
L 2 (Ω)
4 . (2.3)
From the definition of μ ε we conclude that
∫
∇v 2 dx ≥ μ ε ‖v‖ 2 L 2 (Ω)
(2.4)
Ω
for all function v ∈ H 1 (Ω, Γ ε ). Inequalities (2.3) and (2.4) give the following
estimate:
∫
( )∣ v (k)
∣ ∇ + v (l) ∣∣∣
2 (
dx > μ ε 1 − η )
.
2
4
Ω
6
,
If k, l > k 0 (η), it follows that
∫
∣ ∣ ∇v
(k) 2 dx < μ ε + η,
Ω
∫
Ω
∣ ∇v
(l) ∣ ∣ 2 dx < μ ε + η.
Hence,
= 1 2
∫
Ω
≤ μ ε + η
2
∫
Ω
∣ ∇v
(k) ∣ ∣ 2 dx + 1 2
+ μ ε + η
2
( )∣ v (k)
∣ ∇ − v (l) ∣∣∣
2
dx =
2
∫
Ω
∣ ∣ ∫
∇v
(l) 2 dx −
(
− μ ε 1 − η )
4
Ω
(
= η
( )∣ v (k)
∣ ∇ + v (l) ∣∣∣
2
dx
2
1+ μ ε
4
)
→ 0, η → 0.
Finally, according to the Cauchy condition the sequence {v (k) } converges to
some function v ∗ ∈ H 1 (Ω, Γ ε )inthespaceH 1 (Ω, Γ ε ), and
∫
∇v ∗  2 dx = μ ε , ‖v ∗ ‖ 2 L 2 (Ω) =1.
Ω
Assume that v ∈ H 1 (Ω, Γ ε ) is an arbitrary function. Denote
∫
∇(v ∗ + tv) 2 dx
Ω
g(t) =
.
‖v ∗ + tv‖ 2 L 2 (Ω)
The function g(t) is continuously differentiable in some neighborhood of
t = 0. This ratio has the minimum, which is equal to μ ε . Using the Fermat
theorem, we obtain that
∫
2‖v ∗ ‖ 2 L 2 (Ω)
(∇v ∗ , ∇v) dx − 2 ∫ v ∗ vdx ∫ ∇v ∗  2 dx
0=g ′ Ω Ω Ω
 t=0 =
=
‖v ∗ ‖ 4 L 2 (Ω)
∫
∫
=2 (∇v ∗ , ∇v) dx − 2μ ε v ∗ vdx.
Thus,wehaveprovedthat
∫
∫
(∇v ∗ , ∇v) dx = μ ε
Ω
Ω
v ∗ vdx
Ω
Ω
7
for v ∈ H 1 (Ω, Γ ε ), i.e. v ∗ satisfies the integral identity (2.1). This means
that we have found a function v ∗ such that
∫
∫
∇v ∗  2 dx
∇v 2 dx
Ω∫
(v ∗ ) 2 dx = inf Ω
∫
v∈H 1 (Ω,Γ ε)\{0} v 2 dx = μ ε.
Ω
Keeping in mind that u 1 = v ∗ we conclude that μ ε = λ 1 ε . The proof is
complete.
□
Ω
Lemma 2.2. The following Friedrich’s inequality:
∫
∫
U 2 dx ≤ K ε ∇U 2 dx, U ∈ H 1 (Ω, Γ ε ) (2.5)
Ω
holds true, where K ε = 1 .
λ 1 ε
Proof. From Lemma 2.1 we get that
∫
∇U 2 dx
λ 1 Ω
ε ≤ ∫
for any U ∈ H 1 (Ω, Γ
U 2 ε ).
dx
Using this estimate, we deduce that
∫
U 2 dx ≤ 1 ∫
λ 1 ε
Ω
Ω
Ω
Ω
∇U 2 dx.
Denoting by K ε the value 1 , we conclude the statement of our Lemma. □
λ 1 ε
In the following section we will estimate λ 1 ε .
2.2 Auxiliary boundary value problems.
Assume that f ∈ L 2 (Ω) and consider the following boundary value problems:
⎧
⎪⎨ −Δu ε = f in Ω,
u ε =0 onΓ ε ,
(2.6)
⎪⎩ ∂u ε
=0 ∂ν onΓε 2
and {
−Δu 0 = f in Ω,
u 0 =0 on∂Ω.
8
(2.7)
Note that for n = 2 we assume that Γ ε =Γ ε 1 and for n ≥ 3 we assume
that Γ ε =Γ ε 1 ∪ S. The Problem (2.7) is the homogenized (limit as ε → 0)
problem for the Problem (2.6) (a proof of this fact can be found in [5] and
[7], see also [8]).
Consider now the respective spectral problems:
⎧
⎪⎨ −Δu k ε = λk ε uk ε in Ω,
u
⎪⎩
k ε =0 onΓ ε,
∂u k ε
=0 ∂ν onΓε 2
and {
−Δu k 0 = λk 0 uk 0 in Ω,
u k 0 =0 on∂Ω.
Next, let us estimate the difference ∣ 1 − 1
λ
∣ . We will use the method
k ε λ k 0
introduced by O.A.Oleinik, G.A.Yosifian and A.S.Shamaev (see [16], [20]).
Let H ε ,H 0 be separable Hilbert spaces with the inner products (u ε ,v ε ) Hε
and (u, v) H0
,andthenorms‖u ε ‖ Hε and ‖u‖ H0 , respectively; assume that ε is
a small parameter, A ε ∈L(H ε ),A 0 ∈L(H 0 ) are linear continuous operators
and ImA 0 ⊂ V ⊂ H 0 , where V is a linear subspace of H 0 .
C1. There exist linear continuous operators R ε : H ε → H 0 such that for
all f ∈ V we have (R ε f,R ε f) Hε
→ c (f,f) H0
as ε → 0, where c = const > 0
does not depend on f.
C2. The operators A ε ,A 0 are positive, compact and selfadjoint in H ε ,H 0
respectively and sup ‖A ε ‖ L(Hε) < +∞.
ε
C3. For all f ∈ V we have ‖A ε R ε f − R ε A 0 f‖ Hε → 0asε → 0.
C4. The sequence of operators A ε is uniformly compact in the following
sense: if a sequence f ε ∈ H ε is such that sup ε ‖f ε ‖ L(Hε) < +∞, then there exist
a subsequence f ε′ and a vector w 0 ∈ V such that ‖A ε
′f ε′ −R ε
′w 0 ‖ Hε ′
→ 0
as ε ′ → 0.
Assume that the spectral problems for the operators A ε ,A 0 are
A ε u k ε = μk ε uk ε , k =1, 2,..., μ1 ε ≥ μ2 ε
( ) ≥ ...>0,
u
l
ε ,u m ε = δlm ,
A 0 u k 0 = μk 0 uk 0 , k =1, 2,..., μ1 0 ≥ μ2 0
( ) ≥ ...>0,
u
l
0 ,u m 0 = δlm ,
where δ lm is the Kronecker symbol and the eigenvalues μ k ε ,μk 0
according to their multiplicities.
The following theorem holds true (see [16]).
are repeated
9
Theorem 2.1. (OleinikYosifianShamaev) Suppose that the conditions
C1 − C4 are valid. Then μ k ε converges to μk 0 as ε → 0, and the following
estimate
μ k ε − μ k 0≤c − 1 2 sup ‖A ε R ε f − R ε A 0 f‖ Hε (2.8)
f∈N(μ k 0 ,A 0),‖f‖ H0 =1
takes place, where N(μ k 0,A 0 ) = {u ∈ H 0 ,A 0 u = μ k 0u}. Assume also that
k ≥ 1,s≥ 1, are the integer numbers, μ k 0 = ...= μk+s−1 0 and the multiplicity
of μ k 0 is equal to s. Then there exist linear combinations U ε of the eigenfunctions
u k ε ,...,uk+s−1 ε to the Problem (2.6) such that for all w ∈ N(μ k 0 ,A 0)
we get ‖U ε − R ε w‖ Hε → 0 as ε → 0.
To use the method of Oleinik, Yosifian, Shamaev we define the spaces
H ε and H 0 and the operators A ε ,A 0 and R ε in an appropriate way.
Assume that H ε = H 0 = V = L 2 (Ω), and R ε is the identity operator.
The operators A ε ,A 0 are defined in the following way: A ε f = u ε ,A 0 f = u 0 ,
where u ε ,u 0 are the solutions to the Problems (2.6) and (2.7), respectively.
Let us verify the conditions C1 − C4.
The condition C1 is fulfilled automatically because R ε is the identity
operator, c =1. Let us verify the selfadjointness of the operator A ε . Define
A ε f = u ε ,A ε g = v ε ,f,g ∈ L 2 (Ω). Because of the integral identity of the
Problem (2.6) the following identities
∫ ∫
∫
fv ε dx = ∇v ε ∇u ε dx = gu ε dx
Ω
Ω
Ω
are valid.
Hence,
∫
(A ε f,g) L2 (Ω) =(u ε ,g) L2 (Ω) =
∫
u ε gdx=
∇v ε ∇u ε dx =
∫
=
Ω
fv ε dx =(f,v ε ) L2 (Ω) =(f,A ε g) L2 (Ω).
Ω
Ω
The selfadjointness of the operator A 0 can be proved in an analogous way.
It is easy to prove the positiveness of the operator A ε :
∫ ∫
(A ε f,f) L2 (Ω) =(u ε ,f) L2 (Ω) = u ε fdx= ∇u ε  2 dx ≥ 0,
Ω
Ω
10
and ∫ ∇u ε  2 dx > 0iff ≠0. The positiveness of A 0 may be proved in the
Ω
same way.
Next, we prove that A ε ,A 0 are compact operators. Let the sequence {f θ }
be bounded in L 2 (Ω). It is evident that the sequence {A ε f θ } = {u ε,θ } is
bounded in H 1 (Ω, Γ ε ) and the sequence {A 0 f θ } = {u 0,θ } is bounded in
(Ω). Note that {u ε,θ } is bounded uniformly on ε (for a proof see [5]). Because
of compact embedding of the space H 1 (Ω) to L 2 (Ω), we conclude that A ε and
A 0 are compact operators. Moreover,
‖A ε ‖ L(Hε) ≤‖u ε ‖ H 1 (Ω,Γ ε) < M < +∞,
and, consequently,
sup ‖A ε ‖ L(Hε) < M < +∞.
ε
Let us verify the condition C3. The operator R ε is the identity operator
and, thus, it is sufficient to prove that for all f ∈ L 2 (Ω) we have that
‖A ε f − A 0 f‖ L2 (Ω) → 0asε → 0, that is, ‖u ε − u 0 ‖ L2 (Ω) → 0asε → 0.
It is enough to prove that u ε ⇀u 0 in H 1 (Ω). (The week convergence in
H 1 (Ω) gives the strong convergence in L 2 (Ω).) The sequence u ε is uniformly
bounded in H 1 (Ω). Consequently, there exists a subsequence u ε
′, such that
u ′ ε
⇀u ∗ in H 1 (Ω). Further we will set that u ε is the same subsequence. Let
us show that u ∗ ≡ u 0 , that is for all v ∈ H ◦ 1 (Ω)
∫ ∫
fvdx = ∇u ∗ ∇vdx.
◦
H 1
Ω
Ω
The integral identity for the problem (2.6) gives that
∫ ∫
fvdx = ∇u ε ∇vdx.
Ω
Ω
Because u ∗ is a week limit of u ε in H 1 (Ω) the following is valid:
∫
∫
∇u ε ∇vdx→ ∇u ∗ ∇vdx when ε → 0 for all v ∈ H 1 (Ω),
Ω
Ω
and this gives us the desired result, because the integrals ∫ ∇u ∗ ∇vdx and
∫
Ω
fvdx do not depend on ε.
Ω
Let us verify the condition C4. Consider the sequence {f ε }, which is
bounded in L 2 (Ω). Then ‖A ε f ε ‖ H 1 (Ω,Γ ε) = ‖u ε ‖ H 1 (Ω,Γ ε) ≤ const, that is,
11
the sequence {A ε f ε } is compact in L 2 (Ω) and, consequently, there exists a
subsequence ε ′
such that
A ε
′f ε
′
→ w 0 as ε ′ → 0, where w 0 ∈ L 2 (Ω).
Hence, we have that ‖A ε
′f ′ ε
− w 0 ‖ L2 (Ω) → 0asε ′ → 0.
Thus, the conditions C1 − C4 are valid.
It is evident that λ k ε = 1 ,λ k μ k 0 = 1 . Using the estimate (2.8) we have:
ε μ k 0
1
∣ − 1 λ k ε λ k ∣ ≤ sup ‖A ε f − A 0 f‖ H 1 (Ω,Γ ε) =
0 f∈N(λ k 0 ,A 0),‖f‖ L2 (Ω)=1
(2.9)
= sup
f∈N(λ k 0 ,A 0),‖f‖ L2 (Ω)=1
‖u ε − u 0 ‖ H 1 (Ω,Γ ε),
where u ε ,u 0 are the solutions of the Problems (2.6) and (2.7), respectively.
The following inequality
( ) )
1
‖u ε − u 0 ‖ H 1 (Ω,Γ ε) ≤ K‖f‖ L2 (Ω)
((μ ε ) 1 δ
2
2 + , (2.10)
μ ε
∫
∇u 2 dx
Ω
was established in [5], where μ ε = inf ∫ and the constant K
u∈H 1 (Ω,Γ ε 1 )\{0} u 2 dx
Ω
depends only on the domain Ω. Moreover, the following asymptotics was
proved in [5] (the case n = 2) and in [9] (the case n ≥ 3) (see also [1] and
[7]):
{ ( )
π
ln ε
μ ε =
+ O 1
, if n =2;
 ln ε 2
ε n−2 σn c (2.11)
2 ω + O(ε n−1 ), if n>2
as ε → 0.
Here σ n is the area of the unit sphere in R n , and c ω > 0 is the capacity
of the (n − 1)dimensional “disk” ω (see [14] and [18]).
Define the following value:
( ) )
1
ϕ(ε) ≡ K
((μ ε ) 1 δ
2
2 + .
μ ε
)
. Conse
(
Note that if n = 2 it implies that μ ε ∼ 1 and δ = o
ln ε
quently we have:
( )
1
δ o ln ε
ln ε
=
= o(1)
μ ε π
12
1
ln ε
as ε → 0.
In the same way, if n>2 it yields that
δ
∼ o(1)
μ ε
as ε → 0.
Using this asymptotics, we deduce:
⎧
(
⎨(ln ε) − 1 1
2 +(δ(ε) ln ε)
2 + o
)
, if n =2,
(ln ε) − 1 1
2 +(δ(ε) ln ε)
2
ϕ(ε) =K
(
)
⎩ε n 2 −1 +(δ(ε)ε n−2 ) 1 2
+ o ε n 2 −1 +(δ(ε)ε n−2 ) 1 2
, if n>2
(2.12)
as ε → 0.
Finally, due to (2.9), (2.10) and (2.11) we get that
1
∣ − 1 λ 1 ε λ 1 ∣ ≤ ϕ(ε), (2.13)
0
where ϕ(ε) has the asymptotics (2.12).
2.3 Proofs of the main results.
Proof of Theorem 2.1:
Actually, because of estimate (2.5), the Friedrich’s inequality
∫
u 2 dx ≤ 1 ∫
∇u 2 dx, u ∈ H 1 (Ω, Γ
λ 1 ε ) (2.14)
ε
Ω
is valid. The estimate (2.13) implies that
1
≤ 1 + ϕ(ε).
λ 1 0
Ω
λ 1 ε
By rewriting inequality (2.14), using the established relations between λ 1 ε
and λ 1 0 we find that
∫
u 2 dx ≤ 1 ∫
( ) ∫
1
∇u 2 dx ≤ + ϕ(ε) ∇u 2 dx.
λ 1 ε
λ 1 0
Ω
Ω
1
According to our notations, = K
λ 1 0 . Thus, for u ∈ H 1 (Ω, Γ ε ) the following
0
Friedrich’s inequality:
∫
∫
u 2 dx ≤ K ε ∇u 2 dx, K ε = K 0 + ϕ(ε)
Ω
Ω
Ω
13
holds true, where ϕ(ε) ∼ (ln ε) − 1 1
2 +(δ(ε) ln ε)
2 as ε → 0, if n =2. Hence,
the proof is complete.
□
Proof of Theorem 2.2: The proof is completely analogous to that of Theorem
2.1: using inequalities (2.5), (2.12) and (2.13), we obtain the asymptotics
of the constant K ε , hence we leave out the details. Note only that in this
case ϕ(ε) ∼ ε n 2 −1 +(δ(ε)ε n−2 ) 1 2
as ε → 0. □
3 Special cases.
In this section we consider domains with special geometry.
Let ∂G be the boundary of the unit disk G centered at the origin. Assume
that ω ε = {(r, θ) : r =1, −δ(ε) π
Theorem 3.2. Suppose that n = 2, the domain Ω is the unit disk and
δ ln ε −→ 0 as ε → 0. Foru ∈ H 1 (Ω, Γ ε 1 ) the following Friedrich’s inequality
∫
∫
u 2 dx ≤ K ε ∇u 2 dx,
Ω
Ω
K ε = K 0 (1 + 2δ ln sin ε +2δ 2 (ln sin ε) 2 + o(δ 2 ))
as ε → 0, holds true, where K 0 is a constant in the Friedrich’s inequality (0.1)
for functions u ∈ H ◦ 1 (Ω).
Note that nonperiodic geometry are considered in [4]. In the paper the
author constructed and verified the asymptotic expansions of eigenvalues.
Keeping in mind these results it is possible to obtain sharp bounds for the
constant in the Friedrich’s inequality.
Acknowledgments:
The research presented in this paper was initiated at a research stay of Gregory
A. Chechkin at Luleå University of Technology (Luleå, Sweden) in June
2005. The final version was completed when Gregory A. Chechkin was visiting
Laboratoire J.L.Lions de l’Université Pierre et Marie Curie (Paris,
France) in March – April 2007. We thank the referees for several valuable
comments and suggestions, which have improved the final version of the paper.
References
[1] A.G. Belyaev, G.A. Chechkin, and R.R. Gadyl’shin, Effective Membrane
Permeability: Estimates and Low Concentration Asymptotics, SIAM J.
Appl. Math. 60: 1 (2000), 84–108.
[2] D. I. Borisov, Twoparameter asymptotics in a boundaryvalue problem
for the Laplacian, Mathematical Notes 70: 3/4 (2001), 471–485 (Translated
from Mat. Zametki 70: 4 (2001), 520–534).
[3] D. I. Borisov, TwoParametrical Asymptotics for the Eigenevalues of the
Laplacian with Frequent Alternation of Boundary Conditions, Vestnik
Molodyh Uchenyh. Seriya Prikladnaya matematika i mehanika (Journal
of young scientists. Applied mathematics and mechanics) 1 (2002), 36–
52. (in Russian)
15
[4] D. I. Borisov, Asymptotics and estimates for the eigenelements of the
Laplacian with frequently alternating nonperiodic boundary conditions,
Izvestia Mathematics 67: 6 (2003), 1101–1148 (Translated from Izv.
Ross. Akad. Nauk Ser. Mat. 67: 6 (2003), 23–70).
[5] G.A. Chechkin, Averaging of Boundary Value Problems with a Singular
Perturbation of the Boundary Conditions, Russian Academy of
Sciences. Sbornik. Mathematics 79: 1 (1994), 191–222 (Translated from
Mat. Sbornik 184: 6 (1993), 99–150).
[6] G.A. Chechkin, On Spectral Properties of Elliptic Problems with
Rapidly Oscillating Boundary Conditions. In: Boundary Value Problems
for Nonclassical Partial Differential Equations (Ed. Vladimir N.
Vragov), Novosibirsk: Institute of Mathematics, Siberian Division of
the Academy of Sciences of the USSR (IM SOAN SSSR), 1989, 197–200
(in Russian).
[7] G.A.Chechkin, Estimation of Solutions of Boundary–Value Problems
in Domains with Concentrated Masses Located Periodically along the
Boundary: Case of Light Masses, Mathematical Notes 76: 6 (2004),
865–879 (Translated from Matematicheskie Zametki 76: 6 (2004), 928–
944).
[8] G.A.Chechkin, A.L.Piatnitski, and A.S.Shamaev, Homogenization:
Methods and Applications Translations of Mathematical Monographs.
Providence: AMS, 2007 (Translated from Homogenization: Methods and
Applications. Novosibirsk: Tamara Rozhkovskaya Press, 2007 (in Russian)).
[9] R.R. Gadyl’shin and G.A. Chechkin, A Boundary Value Problem for
the Laplacian with Rapidly Changing Type of Boundary Conditions
in a Multidimensional Domain, Siberian Mathematical Journal 40: 2
(1999), 229–244 (Translated from Sibirskii Matematicheskii Zhurnal 40:
2 (1999), 271–287).
[10] R.R. Gadyl’shin, Boundary Value Problems for the Laplacian with
Rapidly Oscillating Boundary Conditions, Doklady Mathematics 58:
2 (1998), 293–296 (Translated from Doklady Akademii Nauk 362: 4
(1998), 456–459).
[11] R.R. Gadyl’shin, On the Eigenvalue Asymptotics for Periodically
Clamped membranes, St. Petersburg Math. J. 10: 1 (1999), 1–14 (Translated
from Algebra i Analiz 10: 1 (1998), 3–19).
16
[12] A. Kufner and L.E Persson, Weighted Inequalities of Hardy Type, World
Scientific, New Jersey–London–Singapore– Hong Kong, 2003.
[13] A. Kufner, L. Maligranda, and L.E. Persson, The Hardy Inequality.
About its History and Related Results, Vydavetelsky Servis Publishing
House, Pilsen, 2007.
[14] N.S.Landkof, Foundations of Modern Potential Theory, SpringerVerlag,
Berlin–New York, 1972.
[15] V. G. Mazja, S. L. Sobolev Spaces, Leningrad, Leningrad Univ. Press,
1985.
[16] O.A. Oleinik, A.S. Shamaev and G.A.Yosifian, Mathematical Problems
in Elasticity and Homogenization, NorthHolland, Amsterdam, 1992.
[17] B. Opic and A. Kufner, Hardy–Type Inequalities, Longman, Harlow,
1990.
[18] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical
Physics, Princeton Univ. Press, Princeton, 1951.
[19] V.S. Vladimirov, Equations of Matematical Physics, “Nauka” Press, oscow,
1976 (in Russian).
[20] G. A. Yosifian, O. A. Oleinik and A. S. Shamaev, On a Limit Behavior
of the Spectrum of a Sequence of Operators Defined in Different Hilbert
Spaces, Russian Mathematical Survey 44: 3 (1989): 195–196 (Translated
from Uspekhi Mat. Nauk 44: 3(267) (1989), 157–158).
G.A.Chechkin,
Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
chechkin@mech.math.msu.su
Yu.O.Koroleva,
Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
korolevajula@mail.ru
17
L.E.Persson,
Department of Mathematics,
Luleå University of Technology,
SE–97187 Luleå, Sweden
larserik@sm.luth.se
18
Paper B
On the Friedrichs Inequality in a Domain
Perforated Aperiodically along the Boundary.
Homogenization Procedure. Asymptotics for
Parabolic Problems 1
G. A. Chechkin, Yu. O. Koroleva,
A. Meidell and L. E. Persson
Abstract
This paper is devoted to the asymptotic analysis of functions depending
on a small parameter characterizing the microinhomogeneous
structure of the domain on which the functions are defined. We derive
the Friedrichs inequality for these functions and prove the convergence
of solutions to corresponding problems posed in a domain perforated
aperiodically along the boundary. Moreover, we use numerical simulation
to illustrate the results.
1 Introduction.
The boundary value problems in the domain perforated along the boundary
were studied by several authors (see, for instance [1], [5], [7], [16], [17]). The
authors considered periodic perforation and random perforation along the
boundary. In these papers and monographs, one can find proofs of the homogenization
theorems, convergence results, and estimates of the rate of
convergence for boundary value problems and for the respective spectral
problems.
In [16, Ch. I, Sec. 3], the method of potentials was used to prove homogenization
theorems for the problem posed in a domain with perforation
along a closed curve. The uniform convergence in absolute value of the solution
of the original problem to a solution of the limit problem in compact
subdomains disjoint from the chosen curve was proved there.
1 The work of the first author was partially supported by RFBR (060100138). The
work of the first and the second authors was partially supported by the program “Leading
Scientific Schools” (HIII1698.2008.1).
1
The papers [15] and [22] are also of interest. Here the problem in a
domain divided by a periodically perforated interior wall is investigated and
the weal convergence of the solution of the original problem to a solution of
the homogenized problem is proved, and, in some cases, estimates for the
rate of convergence are obtained.
In [5], [7] and [19], problems in domains with random structure of perforation
were treated (for details concerning the random structure, see [8] and
[14]). In [5] and [7], the perforation almost surely has the diameter equal to
the distance between holes, whereas, in [19], the perforation disappears in
the limit and gives no contribution to the homogenized problem.
In the present paper, we consider a boundary value problem in a domain
perforated aperiodically along the boundary for the case in which the diameter
of circles and the distance between them are of the same order. Moreover,
we derive the Friedrichs inequality for the functions vanishing on the boundary
of the cavities in the aperiodic case (see the similar paper [6], in which
the function vanishes on small alternating parts of a fixed boundary). It
should be noted that we use a general scheme of proving the convergence,
which was developed in [2], [4], [9] and [20].
This paper is organized as follows. In Section 2, we give geometrical
settings and discuss the main results. In Section 3, we prove the Friedrichs
inequality. Section 4 is devoted to investigations of asymptotics of solutions
to boundary value problems in a domain of the above type, and we also prove
the convergence of solutions, as well as the convergence of the corresponding
eigenelements, to those for the problem in a domain perforated aperiodically
along the boundary. In Section 5, we apply numerical simulations for the
parabolic problem in a domain of the above type with inhomogeneous structure
along the boundary. In the appendix, we prove an auxiliary result that
we need concerning properties of solutions to boundary value problems in a
domain perforated aperiodically along the boundary.
2 Geometrical settings.
Denote by Ω the rectangle [0,a] × [0,b]. Assume that ε ≪ 1isasmall
parameter, and introduce another small parameter by the rule ε i 1 = α i ε,
α i = const, 0< A
where B = const, i ∈ N. Denote by B ε the union ∪ i B i ε. Define the domain
Ω ε =Ω\ B ε (see Fig.1).
Figure 1: Domain perforated along the boundary.
We use the following notations:
Γ 1 = {(x, y) :y = b, 0 ≤ x ≤ a},
Γ 3 = {(x, y) :x = a, 0 ≤ y ≤ b},
Γ 2 = {(x, y) :x =0, 0 ≤ y ≤ b},
Γ 4 = {(x, y) :y =0, 0 ≤ x ≤ a}.
Thus, ∂Ω =Γ 1 ∪ Γ 2 ∪ Γ 3 ∪ Γ 4 and Γ ε = ∂B ε .
Denote by H 1 (Ω ε , Γ ε ) the set of functions in H 1 (Ω ε ) with zero trace on Γ ε .
Similarly, by H 1 (Ω, Γ 4 ) we denote the set of functions in H 1 (Ω) with zero
trace on Γ 4 . We also use the space C ∞ 0 (Ω, Γ 4) which consists of the functions
in C ∞ (Ω), vanishing in the neighborhood of Γ 4 . Analogously, C ∞ 0 (Ω ε, Γ ε )is
the set of functions from C ∞ (Ω ε ), vanishing in the neighborhood of Γ ε .
Remark 1. Further, we identify functions belonging to H 1 (Ω ε , Γ ε ) and functions
in H 1 (Ω) vanishing on B ε . Analogously, we identify functions belonging
to L 2 (Ω ε ) and functions from L 2 (Ω) which equal to zero in B ε . We use the
same notation for functions in L 2 (Ω) restricted to Ω ε .
Our aim is to derive the Friedrichs inequality for functions u ε ∈ H 1 (Ω ε , Γ ε )
and use it to study the behavior of solutions to an elliptic problem and to a
related parabolic problem.
3 The Friedrichs inequality.
In this section, we derive the Friedrichs inequality for functions belonging to
H 1 (Ω ε , Γ ε ).
3
Theorem 3.1. The inequality
∫
∫
u 2 ε (x, y) dx dy ≤M
Ω ε
Ω ε
∇u ε (x, y) 2 dx dy (3.1)
holds true for any functions u ε ∈ H 1 (Ω ε , Γ ε ), where the constant M does not
depend on ε.
Proof. Let us represent the domain Ω ε in the form
Π i+
ε 1
∪ Π i−
ε 1
∪ Π i ε 2
,
Ω ε = ⋃ i
where Π i+
ε 1
is the domain of width 2ε i 1 bounded from below by the ith upper
semicircle and bounded from above by the line y = b, Π i−
ε 1
is the domain of
width 2ε i 1 bounded from above by the i lower semicircle and bounded from
below by the line y = 0, and, finally, Π i ε 2
is the rectangle of width ε i 2 −εi 1 −εi+1 1 ,
if ε i 2 >εi 1 + εi+1 1 (and of width 0 if ε i 2 = εi 1 + εi+1 1 ) and of height b (see Fig.2).
Figure 2: The division of the domain Ω ε .
Denote by Π ω,i+
ε 1
the domain {(x, y) ∈ Π i+
ε 1
,y
Figure 3: The division of the layer
since u ε (x, y 0 )=0.
By the CauchySchwartzBunyakovsky inequality,
⎛
∫ ω
u 2 ε (x, ω) = ⎝
y 0
⎞2
∫
∂u ω
ε
∂y dy ⎠ ≤ ω
y 0
( ) 2 ∫ ω
∂uε
dy ≤ ω
∂y
∇u ε  2 dy. (3.2)
y 0
In this case, integrating inequality (3.2) with respect to x over Γ ω,i
ε 1
∫
∫
u 2 ε (x, ω) dx ≤ ω
Γ ω,i
ε 1
It follows from (3.3) that
∫
∫
u 2 ε(x, ω) dx ≤ b
Γ ω,i
ε 1
we obtain
Π ω,i+
ε 1
∇u ε  2 dx dy. (3.3)
Π i+
ε 1
∇u ε  2 dx dy. (3.4)
Finally, integrating (3.4) with respect to ω over [ε, b], we see that
∫
∫
u 2 ε dx dy ≤ b2
Π i+
ε 1
Π i+
ε 1
∇u ε  2 dx dy. (3.5)
5
The case in which (x, ω) ∈ Π i−
ε 1
,ω ε(the case
in which ω
Squaring the formula and using the CauchySchwartzBunyakovsky inequality,
we obtain the following inequality
u 2 ε (x, ω) ≤ √ (x,ω)
∫ ( ) 2 ∫
∂uε
(ω − ε) 2 +(α i + β i ) 2 ε 2 dl ≤ c i ω
∂l
(˜x,ỹ)
(x,ω)
(˜x,ỹ)
∇u ε  2 dl,
where the constant c i does not depend on ε. Keeping in mind the estimates
for α i and β i , we conclude that there exists a constant C 1 such that c i
to the corresponding eigenelements for the limit problem
⎧
⎪⎨ −Δu 0 = λ 0 u 0 in Ω,
u 0 =0 onΓ 4 ,
⎪⎩ ∂u 0
=0 onΓ ∂ν 1 ∪ Γ 2 ∪ Γ 3 ,
(4.2)
as ε → 0, as well as the convergence of solutions to the boundary value
problem ⎧
⎪⎨ −ΔU ε = λU ε + F in Ω ε ,
U
⎪ ε =0
onΓ ε
(4.3)
⎩ ∂U ε
=0 on∂Ω, ∂ν
to the solution of the following homogenized problem:
⎧
⎪⎨ −ΔU 0 = λU 0 + F in Ω,
U 0 =0 onΓ 4 ,
(4.4)
⎪⎩ ∂U 0
=0 onΓ ∂ν 1 ∪ Γ 2 ∪ Γ 3 ,
as ε → 0.
This means that the problem (4.2) is the limit (homogenized) problem
for (4.1) and (4.4) is the limit (homogenized) problem for (4.3).
Denote by (·, ·) 0 and (·, ·) 1 the inner products in spaces L 2 (Ω) and H 1 (Ω)
corresponding to the norms ‖·‖ 0 and ‖·‖ 1 , respectively.
Further, denote by (X, Y) the inner product of vectors X and Y. Consider
weak solutions to problems (4.1), (4.2), (4.3) and (4.4) (see, for instance,
[23]). The function U ε ∈ H 1 (Ω ε , Γ ε ) is a weak solution to the boundary value
problem (4.3) if the following integral identity
∫
∫
∫
(∇U ε , ∇v) dx dy − λ U ε vdxdy= F vdxdy (4.5)
Ω ε
Ω ε
Ω ε
holds for any v ∈ H 1 (Ω ε , Γ ε ); a function U 0 ∈ H 1 (Ω, Γ 4 ) is a referred to as a
weak solution to the boundary value problem (4.4) if the integral identity
∫
∫
∫
(∇U 0 , ∇v) dx dy − λ U 0 vdxdy= F vdxdy (4.6)
Ω
Ω
is valid for any v ∈ H 1 (Ω, Γ 4 ). Similarly, a function u ε ∈ H 1 (Ω ε , Γ ε )isan
eigenfunction for the spectral problem (4.1) corresponding to an eigenvalue
λ ε , if the integral identity
∫
∫
(∇u ε , ∇v) dx dy = λ ε u ε vdxdy
Ω
Ω ε
Ω ε
8
holds for any v ∈ H 1 (Ω ε , Γ ε ); finally, a function u 0 ∈ H 1 (Ω, Γ 4 )isaneigenfunction
for the spectral problem (4.2) corresponding to an eigenvalue λ 0 if
the following integral identity
∫
∫
(∇u 0 , ∇v) dx dy = λ 0 u 0 vdxdy
Ω
is valid for any v ∈ H 1 (Ω, Γ 4 ).
Note that, due to Remark 1, the integral identity (4.5) can be represented
as follows:
∫
∫
∫
(∇U ε , ∇v) dx dy − λ U ε vdxdy= F vdxdy. (4.7)
Ω
The main goal of this section is to prove the following theorems.
Ω
Theorem 4.1. Assume that F ∈ L 2 (Ω) and K is an arbitrary compact set in
the complex plane C such that K contains no eigenvalues of the limit problem
(4.2). In this case, the following statements hold.
1. There is a number ε 0 > 0 such that, for any ε
The proof of these theorems uses the approach developed in [2], [4], [11],
[12],[13] and [20] for divers types of singular perturbations. However, we must
modify this approach to fit it into the geometric structure of our domain. Let
us now prove some auxiliary statements which are needed in our analysis.
The proof of the following Lemma is similar to that of the convergence
theorem in [3] and [4].
Lemma 4.1. Let v ε be a sequence of functions from H 1 (Ω ε , Γ ε ) and assume
that v ε ⇀v ∗ weakly in H 1 (Ω) as ε → 0. Then v ∗ ∈ H 1 (Ω, Γ 4 ).
Proof. Introduce the following sets:
Γ ω =Ω ε
⋂
{(x, y) :y = ω}, i.e. Γ ω = ⋃ i
(
Γ
ω,i
ε 1
∪ Γ ω,i
ε 2
)
Π ω =Ω ε
⋂
{(x, y) :y
and centered at the center of B j ε. Write Ω 1 ε =Ω\ K ε . Introduce the following
cutoff functions: ψ j (s) ∈ C ∞ (R), 0 ≤ ψ j ≤ 1, which vanishes for s ≤ α j ε,
and equals to 1 as s ≥ r j ε, and
Ψ ε (x, y) = ∏ i
⎛
ψ i
⎝ 1 √ (x − ε
ε
⎞
i∑
β j ) 2 +(y − ε) 2 ⎠ . (4.13)
j=0
We can readily see that Ψ ε =0inB ε and Ψ ε =1inΩ 1 ε .
The following lemma is used in our further analysis.
Lemma 4.2. Suppose that v ∈ H 1 (Ω, Γ 4 ). Then the sequence vΨ ε converges
to the function v strongly in H 1 (Ω ε ).
Proof. We must prove the convergence
‖vΨ ε ‖ 2 1 →‖v‖ 2 1 as ε → 0 (4.14)
to prove the strong convergence vΨ ε to the function v in H 1 (Ω ε ).
First assume, that v ∈ C ∞ 0 (Ω, Γ 4). Then
∫
∫
∫
‖vΨ ε ‖ 2 1 = v 2 Ψ 2 ε dx dy + ∇v 2 Ψ 2 ε dx dy +
Ω ε Ω ε
∫
+2
Ψ ε v(∇v, ∇Ψ ε ) dx dy.
Ω ε
v 2 ∇Ψ ε  2 dx dy+
Ω ε
Due to the properties of the function Ψ ε , it is clear that
∫
Ω
∫
Ω ε
∫
Ω
∫
Ω ε
v 2 Ψ 2 ε dx dy + ∇v 2 Ψ 2 ε dx dy → v 2 dx dy +
ε ε
∇v 2 dx dy = ‖v‖ 2 1
(4.15)
as ε → 0.
We claim that ∫ v 2 ∇Ψ ε  2 dx dy tends to zero as ε → 0. Taking into
Ω ε
account the properties of function Ψ ε , we obtain
∫
v 2 ∇Ψ ε  2 dx dy = ⋃ ∫
v 2 ∇ψ i  2 dx dy. (4.16)
i
Ω ε
K i ε \Bi ε
11
Let us estimate ∇ψ i  2 . Obviously,
∂ψ i
∂x =
∑
(x − ε i β j )
j=0 dψ i
√
∑
ε (x − ε i ds ,
β j ) 2 +(y − ε) 2
j=0
∂ψ i
∂y = (y − ε) dψ
√
i
∑
ε (x − ε i ds .
β j ) 2 +(y − ε) 2
j=0
Therefore,
∇ψ i  2 ≤ C 4
ε as (x, y) ∈ 2 Ki ε \ Bi ε . (4.17)
Let Ξ ε := {(x, y) ∈ Ω ε :0
where C 7 does not depend on ε and λ. It is clear that formula (4.20) follows
from this inequality.
1. The existence and uniqueness of a solution to Problem (4.3) follows
from the estimate (4.8).
Indeed, the integral identity (4.7) can be represented as follows :
∫
∫
∫
∫
(∇U ε , ∇v) dx dy+ U ε vdxdy−(λ+1) U ε vdxdy= F vdxdy. (4.21)
Ω
Ω
Ω
Ω
This means that identity (4.21) is equivalent to the relation
(U ε ,v) 1 − (λ +1)(U ε ,v) 0 =(F, v) 0 . (4.22)
The Riesz theorem (see, for example, [26]) says that there is an operator A
for which (U ε ,v) 0 =(AU ε ,v) 1 . Hence, identity (4.22) is equivalent to
(U ε ,v) 1 − (λ +1)(AU ε ,v) 1 =(F, v) 0 .
The equation is solvable because the kernel of the operator [I − (λ +1)A]
is trivial (since the solution to problem (4.3) is unique; the uniqueness of
solutions to (4.3) follows from (4.8)).
Now let us derive now estimate (4.8). Assume that estimate (4.8) fails to
hold, i.e., there exist sequences ε k ,F k and λ k such that ε k → 0ask →∞,
F k ∈ L 2 (Ω) and the inequality
‖U εk ‖ 1 >k‖F k ‖ 0 (4.23)
holds for the solutions to problem (4.3) with ε = ε k ,λ= λ k ,F= F k .
Without loss of generality, we may assume that the sequence U ε is normalized
in L 2 (Ω), i.e.
‖U εk ‖ 0 =1. (4.24)
(If ‖U εk ‖ 0 ≠1, then, dividing (4.20) by ‖U εk ‖ 0 and writing V εk = Uε k
‖U εk ‖ 0
,
we obtain the similar inequality
(
‖V εk ‖ 1
≤ C 9 ‖V εk ‖ 0
+ 1 )
( )
‖F ‖
‖U εk ‖ 0
≤ C 10 ‖Vεk ‖ 0
+ ‖F ‖ 0 ,
0
( )
C
where ‖V εk ‖ 0 =1. The value max C 9 , 9
‖U εk ‖ 0
can be taken for C 10 . Hence,
we can assume that ‖U εk ‖ 0 = 1 from the very beginning.)
Now using (4.20) and (4.23), we obtain
13
‖U εk ‖ 1 ≤ C 11 , ‖F k ‖ 0 < C 11
as k →∞. (4.25)
k
As is known, a bounded set of H 1 is weakly compact in H 1 , and, hence
there exists a subsequence {k ′ } of indexes {k} and U ∗ and λ ∗ such that
λ k ′ → λ ∗ ∈ K, U εk ′
⇀U ∗ in H 1 (Ω) as k ′ → +∞. (4.26)
The inequalities (4.24) and (4.26) give
U ∗ ≠0.
Suppose that v is an arbitrary fixed function in C0 ∞(Ω, Γ 4).
Then v ∈ H 1 (Ω ε , Γ ε ) for all small ε. It is clear that vΨ ε ∈ H 1 (Ω ε , Γ ε ), where
the function Ψ ε is defined in (4.13). Substitute vΨ εk ′
as a test function,
U = U εk ′
,λ = λ k ′ and F = F k ′ in the integral identity (4.7). We obtain
∫
(
∇Uεk
′
, ∇(vΨ εk ′) ) ∫
∫
dx dy − λ k ′ U εk ′vΨ εk ′
dx dy = F k ′vΨ εk ′
dx dy.
Ω
Ω
(4.27)
Passing to the limit in the integral identity (4.27) as ε k ′ → 0 and using (4.25),
(4.26) and Lemma 4.2, we obtain the integral identity
∫
∫
(∇U ∗ , ∇v) dx dy = λ ∗ U ∗ vdxdy.
Ω
Ω
Ω
It follows from the density of the embedding of C0 ∞(Ω, Γ 4)inH 1 (Ω, Γ 4 )
that this identity remains valid for any v ∈ H 1 (Ω, Γ 4 ). It follows from Lemma
4.1 that U ∗ ∈ H 1 (Ω, Γ 4 ). Since U ∗ ≠0andv is an arbitrary function in
H 1 (Ω, Γ 4 ), it follows that λ ∗ ∈ K is an eigenvalue of the limit problem (4.2).
However, we assumed that K contains no eigenvalues of the limit problem
(4.2). This contradiction proves estimate (4.8).
2. Let λ ∈ K be an arbitrary fixed number and assume that the sequence
{ε k }→0ask → +∞. We can choose a subsequence weakly convergent in
H 1 from a sequence bounded in H 1 . Hence, there is a subsequence {k ′ } of
indexes {k} and a function U ∗ such that U εk ′
⇀U ∗ in H 1 (Ω) (and strongly
in L 2 (Ω)) as k ′ → +∞.
Recalling Lemma 4.1, we see that the limit function U ∗ is in H 1 (Ω, Γ 4 ).
Passing to the limit in the integral identity (4.27)( using the same reasoning
as in the proof of 1 part of the theorem), we have
∫
∫
∫
(∇U ∗ , ∇v) dx dy = λ ∗ U ∗ vdxdy+ F vdxdy,
Ω
Ω
14
Ω
which coincides with the integral identity of the problem (4.4). Since the
solution to problem (4.4) is a unique, we conclude that U ∗ = U 0 . Moreover,
it follows from (4.26) that
U ε → U 0 strongly in L 2 (Ω) and weakly in H 1 (Ω) as ε → 0.
Substituting v = U εk in the integral identity (4.21) and using the convergence
of the function in L 2 (Ω), we obtain
‖∇U ε ‖ 2 0 −‖∇U 0‖ 2 0 = ∫
∫
FU ε dx dy −
FU 0 dx dy =
Ω
∫
=
Ω
Ω
F (U ε − U 0 ) dx dy −→ 0 as ε → 0.
(4.28)
From (4.28) and the weak convergence of U εk to U 0 in H 1 (Ω) imply the
strong convergence in this space. Hence, the convergence (4.9) holds, because
ε k is an arbitrary sequence. The theorem is proved.
The following lemma can be proved by using the lines of the proof of
assertion 2) of Theorem 4.1.
Lemma 4.3. Assume that λ ε is an eigenvalue of problem (4.1) which converges
to an eigenvalue λ 0 of the limit problem (4.2) as ε → 0. Let u ε be an
eigenfunction corresponding to the eigenvalue λ ε and normalized in L 2 (Ω).
Then the sequence {ε k } ∞ k=1 → 0 admits a subsequence {ε k m
} ∞ m=1 → 0 such
that
‖u εkm − u 0 ‖ 1 → 0,
where u 0 is an eigenfunction corresponding to the eigenvalue λ 0 and normalized
in L 2 (Ω).
The mutual orthogonality of the eigenfunctions of problem (4.2), a similar
fact for problem (4.1) and Lemma 4.3 give the following statement.
Lemma 4.4. Let the multiplicity of an eigenvalue λ 0 for problem (4.2) be
equal to N. Suppose that the eigenvalues λ ε of problem (4.1) converge to λ 0 .
Then the multiplicity of λ ε is less then or equal to N.
Below use the following lemma, which is proved in the appendix.
15
Lemma 4.5. Let U 0 ,U ε be solutions to problems (4.3) and (4.4), respectively.
Assume that K is an arbitrary compact set on the complex plane C and K
contains no eigenvalues of the spectral problem (4.2). Then the following
representations hold:
∞∑ (F, u n 0
U 0 =
) 0
λ n 0 − λ un 0,
U ε =
n=1
∞∑
n=1
(F, u n ε ) 0
λ n ε − λ un ε ,
where u n 0 ,un ε are the eigenfunctions and λn 0 ,λn ε
problems (4.2) and (4.1), respectively.
Proof of Theorem 4.2. Let
be eigenvalues of problem (4.2), let
0
where u p+i
∗ stand for the eigenfunctions of problem (4.2) which are orthogonal
and normalized in L 2 (Ω). Without loss of generality, we assume that
u p+i
∗
= u p+i
0 , i.e., that the following convergence relation holds:
‖u p+i
ε km
− u p+i
0 ‖ 0 → 0, i =1,...,M. (4.31)
Define
F := u p+M+1
0 ,
G 0 (λ) :=(U 0 (·,λ),u p+M+1
0 ) 0 ,
G ε (λ) :=(U ε (·,λ),u p+M+1
0 ) 0 ,
where U 0 and U ε are the solutions to problems (4.3) and (4.4), respectively.
Let S(t, z) be a circle of radius t centered at the point z on the complex
plane. There is a T>0 such that the circle S(t, λ 0 ) contains no eigenvalues
of the limit problem except for λ 0 whenever t
there is another subsequence for which these eigenvalues converge to some
λ ∗ ∈ S(T,λ 0 ), λ ∗ ≠ λ 0 , but this leads to a contradiction, since the circle
S(T,λ 0 ) contains no eigenvalues of problem (4.2) except λ 0 .Consequently,
∑ ∞
g ε (λ) ≤(T − t) −1 (u p+M+1
0 ,u s ε km
) 0  2 . (4.36)
s=1
Since the system {u s ε km
} ∞ s=1 forms an orthogonal basis in L 2 (Ω ε ), the Parseval
Steklov formula (see, for instance, [25]):
‖F ‖ 2 L 2 (Ω = ∑ ∞
ε) F i  2 , F i =(F, u i ε km
) 0
i=1
holds for any F ∈ L 2 (Ω). It follows from (4.36) and from the ParsevalSteklov
formula that
g ε (λ) ≤(T − t) −1 ‖u p+M+1
0 ‖ 0 =(T − t) −1 ,
i.e., the series (4.35) converges uniformly with respect to λ ∈ S(t, λ 0 ). Hence,
by Weierstrass’ theorem g ε is a holomorphic function of λ ∈ S(t, λ 0 ).
Consequently, using (4.32),(4.34) and the Lebesgue dominated convergence
theorem, we obtain
∫
∫
G εkm (λ)dλ −→ G 0 (λ)dλ, m →∞ (4.37)
∂S(t,λ 0 )
∂S(t,λ 0 )
for t
Remark 2. Using Theorems 4.1 and 4.2, we conclude that the following
Friedrichs inequalities with sharp constants hold true for u ε ∈ H 1 (Ω ε , Γ ε )
and u 0 ∈ H 1 (Ω, Γ 4 ):
∫
∫
u 2 ε dx dy ≤M ε ∇u ε  2 dx dy
Ω ε
Ω ε
∫
∫
u 2 0 dx dy ≤M 0
∇u 0  2 dx dy,
and
Ω
M ε −M 0 →0 as ε → 0.
This follows from the relations M ε = 1 and M
λ 1 0 = 1 , which can be proved
ε
λ 1 0
by using the variational principle for problems (4.1) and (4.2), respectively.
Remark 3. Results similar to those obtained in this section and in the previous
ones can also be obtained if Ω ε is a domain with sufficiently smooth
boundary perforated aperiodically along the boundary (see Fig. 5).
Ω
Figure 5: The domain Ω ε .
5 Numerical simulations for the respective
parabolic problem.
In the previous section, we considered the elliptic boundary value problem in
a domain which is perforated aperiodically along the boundary. This section
is devoted to numerical studying the asymptotic behavior of solutions to the
corresponding parabolic problem.
19
Consider the following heat problem
⎧
∂v ε
= σΔv
∂t ε in Ω ε ,
⎪⎨
v ε =0 onΓ ε ,
∂v ε
=0 on∂Ω,
∂ν ⎪⎩
v ε (·, 0) = g(·) in Ω ε ,
and the corresponding limit problem
⎧
∂v 0
= σΔv
∂t 0 in Ω,
⎪⎨
v 0 =0 onΓ 4 ,
∂v 0
∂ν ⎪⎩
=0 onΓ 1 ∪ Γ 2 ∪ Γ 3 ,
v 0 (·, 0) = g(·) in Ω,
(5.1)
(5.2)
where g stands for a smooth function and σ for the thermal diffusivity. The
fact that problem (5.2) is the homogenized (limit) for problem (5.1) can be
verified in just the same way as in Section 4. In fact, it can be shown that
the solutions v ε and v 0 are the limits (or at least sublimits) of sequences of
the form
M∑
v ε,M (x, t) = u i ε (x)di ε (t),
and
v 0,M (x, t) =
i=1
M∑
i=1
u i 0(x)d i 0(t),
respectively, as M → +∞. Here, u i ε and v0 i are eigenfunctions of problems
(5.1) and (5.2), respectively, and v ε,M (·,t)andv 0,M (·,t) are the projections
of the solutions v ε (·,t)andv 0 (·,t) to the finitedimensional subspace spanned
by the eigenfunctions {u i ε} M i=1 and {ui 0} M i=1
, respectively (see e.g. [10]). Using
convergence results for the eigenvalues, one can prove that v ε (x, t) converges
(in an appropriate norm) to v 0 (x, t) ateverychosenpointx ∈ Ω. Alternatively,
one can use Rothe’s method of discretization in time (see, for instance,
[21]) to prove the convergence of solutions of the problem (5.1) to the solution
of the problem (5.2).
As an illustration of our theoretical results, we can use numerical simulations.
The asymptotic behavior of the solutions to a parabolic problem must
be similar to the behavior of solutions to the corresponding elliptic problem.
As an example, we consider the case with σ =1,g(x) = 1 for all x ∈ Ω,
a = b = 10, ε =1/2, ε i 1 =(2/3) and ε =1/3,εi 2 =2ε =1. In this case, the
20
Figure 6: The calculated temperature distribution obtained by using the
finite element program ANSYS 11 for t =2.5
solution of the limit problem becomes
∞∑
v ε (x, t) =
k=0
(
(2k+1)
e
−(
20
π) 2 t 4
(2k +1)π
( ( )))
(2k +1)
sin πx 2
20
In Figure 6, we can see the computed temperature distribution obtained
by using the finiteelement program ANSYS 11 for t =2.5. It turns out that
the temperature at the upper surface is almost constant in the horizontal direction
at any time. The curves of the temperatures v ε and v 0 at that surface
are presented in Fig. 7 and Fig. 8, respectively. We observe immediately
that these curves are relatively close to each other (as we would expect from
our theoretical results).
Thus, using different approaches (homogenization theory, method of integral
inequalities and estimates, and numerical simulation), we have obtained
the same results.
Appendix.
In this section we prove Lemma 4.5. It should be noted that the sketch of
the proof first appeared in [2]. Here we use the same scheme; however, we
21
Figure 7: The curve of the temperature v ε at the upper surface as function
of t.
Figure 8: The curve of the temperature v 0 at the upper surface (as function
of t).
22
modify it according to our microinhomogeneous structure.
Proof of Lemma 4.5. As is well known (see, for instance, [25]), the formula
⎛
⎞
∫
∫
(f,g) 1,0 := (∇f,∇g)dx ⎝(f,g) 1,ε := (∇f,∇g)dx⎠
Ω
Ω ε
defines an inner product on the space H 1 (Ω, Γ 4 )(onthespaceH 1 (Ω ε , Γ ε ))
corresponding to the norm ‖f‖ 2 1,0 =(f,f) 1,0 (‖f‖ 2 1,ε =(f,f) 1,ε), which is
equivalent to the norm on H 1 (Ω) (on H 1 (Ω ε )). As is also known (see, for
instance, [25]) the systems systems
u 1 δ u s δ
√ ,..., √ ,..., δ ≥ 0 (5.3)
λ
1
δ
λ
s
δ
form orthogonal and normalized basis in H 1 (Ω, Γ 4 )forδ =0andinH 1 (Ω ε , Γ ε )
for δ = ε>0. Hence, the function U δ can be expanded in the Fourier series
with respect to the orthogonal basis (5.3) as follows:
U δ =
∞∑
s=1
(U δ ,u s δ ) 1,δ
u s
λ s δ = ∑ ∞
δ
s=1
(U δ ,u s δ ) 1,0
u s
λ s δ , δ ≥ 0 (5.4)
δ
in H 1 (Ω, Γ 4 ) for any function U 0 ∈ H 1 (Ω, Γ 4 )ifδ =0andinH 1 (Ω ε , Γ ε )for
any function U ε ∈ H 1 (Ω ε , Γ ε )ifδ = ε>0. The integral identities (4.6) and
(4.7) lead to the following formula
(U δ ,u s δ ) 1,0 = λ(U δ ,u s δ ) 0 +(F, u s δ ) 1,0.
Substituting the expansion (5.4) into the righthand side of this equation, we
obtain
(U δ ,u s δ) 1,0 = λ (U
λ s δ ,u s δ) 1,0 +(F, u s δ) 0 .
0
Hence,
(U δ ,u s δ ) 1,0 = λs 0
λ s 0 − λ(F, us δ ) 0.
Substituting this formula into (5.4), we obtain the representations (4.33).
The lemma is proved.
23
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G.A.Chechkin,
Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
&
Narvik University College
Postboks 385, 8505 Narvik, Norway,
chechkin@mech.math.msu.su
Yu.O.Koroleva,
Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
&
Department of Mathematics,
Luleå University of Technology,
26
SE–97187 Luleå, Sweden
korolevajula@mail.ru
A.Meidell,
Narvik University College
Postboks 385, 8505 Narvik, Norway,
Annette.Meidell@hin.no
L.E.Persson,
Department of Mathematics,
Luleå University of Technology,
SE–97187 Luleå, Sweden
&
Narvik University College
Postboks 385, 8505 Narvik, Norway,
larserik@sm.luth.se
Paper C
On the Friedrichs inequality in a cube
perforated periodically along the part of the
boundary. Homogenization procedure
Yu.O. Koroleva
Key words and phrases.Inequalities, Partial Differential Equations, Functional
Analysis, Spectral Theory, Homogenization Theory, Friedrichstype
Inequalities
AMS 2009 Subject Classification. 39A10, 39A11, 39A70,
39B62, 41A44, 45A05
Abstract
This paper is devoted to the application of asymptotic analysis for
functions depending on small parameter which characterizes the microinhomogeneous
structure of the domain where the functions are
defined. We derive the Friedrichs inequality for functions set in the
three dimensional domain perforated periodically along a part of the
boundary and prove the convergence of solutions of the original problems
to the solution of the respective homogenized problem in this
domain.
1 Introduction
There exist a lot of literature, where boundaryvalue problems in perforated
domains are studied (see e.g. [2][4], [8][15], [26], [27], [31][37], [44][51]). In
these papers and monographs the authors studied different kinds of perforation
for linear as well as for nonlinear differential operators.
First we discuss some results concerning boundaryvalue problems for
linear partial differential equations in perforated domains.
Boundaryvalue problems in perforated domains were studied in works of
O. A. Oleinik and her coauthors (see [24], [25], [42], [43] and [44]) in the case
when cavities are distributed periodically with a small period provided that
the sizes of the cavities have the same order as the period of perforation. The
case when the sizes of the cavities are smaller than the period of perforation
was considered in [13], [26] and [27].
The work [3] (see also the short note [4]) was dedicated to analysis of
boundaryvalue problems in perforated domains in the case when the diameter
of the holes is smaller than the distance between them. In particular,
a homogenization theorem was proved for a boundaryvalue problem with
1
Neumann boundary conditions on the external boundary and with Dirichlet
boundary conditions on the boundary of cavities. Moreover, the author
constructed the first term of the asymptotic expansion of the solutions of
the original problems and obtained some estimates of the difference between
the solutions of the original problems and the first term of the asymptotic
expansion. The author studied also spectral properties of the corresponding
spectral problems and obtained some estimates of the difference between the
eigenelements of the original problems and the corresponding eigenelements
of the homogenized problem.
In [37] the Dirichlet problem was considered in a domain with aperiodically
distributed cavities. In particular, the weak convergence in L 2 of the
solutions was proved in terms of convergence of the harmonic capacities of
the cavities.
Boundaryvalue problems in domains with perforation along a surface
were studied in [15], [33], [34, Ch. I, §3] and [47].
In [34, Ch. I, §3] the author studied a boundaryvalue problem in a
domain perforated along a closed curve. In particular, the uniform convergence
of the solutions of the original problems to the corresponding solution
of the limit problem was proved for compact subdomains, containing the
curve. In [47] a boundaryvalue problem was stated in a domain divided by
a perforated wall. In particular, the weak convergence of the solutions of
the original problems to the solution of the limit problem was established.
Moreover, the estimates of the rate of the convergence were obtained for
some special cases. In [15] the author considered the thick Neumann sieve
problem, which is an extension of a problem raised by V.O. Marchenko and
E. Ya. Khruslov (see [34]) and, more specifically, by E. SánchesPalencia (see
[47]): the plane x n =0, perforated by εperiodically distributed small holes
with diameters r(ε) < ε , separates an ndimensional bounded domain G into
2
two subdomains G + and G − . The behavior of the solutions u ε of boundaryvalue
problems in G was studied when ε → 0+. In [15] an analogous problem
was considered for a “thick” sieve enlarged to thickness h(ε) with holes being
cylinders of the height h(ε).
In [33] some homogenization theorems were obtained for solutions of
boundaryvalue problems with different types of conditions on the boundary
of cavities. In particular, the convergence of the solutions of the original
problems to the solution of the corresponding homogenized problem was
proved. Moreover, some estimates of the rate of the convergence for the solutions
as well as for the eigenelements of the corresponding spectral problems
were obtained. Besides this, the same boundaryvalue problems were studied
in a domain divided by a perforated wall into two subdomains. In particular,
the authors proved that in the case of mixed boundary conditions or
2
Dirichlet condition the limit problem has a jump of the boundary conditions
on the wall under the assumption that the size of perforation is sufficiently
small. If the diameters of small sets is too “large”, then the Dirichlet boundary
condition vanishes on the wall in the limit. The authors obtained also
some estimates of the rate of the convergence of the solutions as well as of
eigenelements for these boundaryvalue problems.
Boundaryvalue problems in perforated domains for nonlinear differential
operators were considered e.g. in [31], [32], [48], [49], [50] and [51]. The
monograph [48] deals with boundaryvalue problems for nonlinear elliptic
equations of an arbitrary order. The first half of the book deals with the
solvability problem while the second one is concerned with properties of the
weak solutions. Moreover, the author considers homogenization of a family of
quasilinear boundaryvalue problems both in domains with the finegrained
boundary and in domains with channels. In particular, it was proved that
solutions of these boundaryvalue problems are close to the solution of the
corresponding limit quasilinear problem in the corresponding nonperforated
domain. In [48], [49] and [50] Dirichlet boundaryvalue problems were considered
in perforated domains for nonlinear equations. Moreover, further
investigations of quasilinear boundaryvalue problems in a sequence of domains
with finegrained boundaries can be found in [51]. In particular, the
authors study convergence of eigenvalues of nonlinear Dirichlet problems in
such domains. Here the domain as follows was considered:
Ω s =Ω\
I(s)
⋃
i=1
F (s)
i ,
where Ω ⊂ R n is an arbitrary domain and F (s)
i ,i=1,...I(s) < ∞, s∈ N is
a finite number of disjoint closed domains, which are contained in Ω. Here it
is also assumed that the diameters of the sets F (s)
i , the distance between two
neighboring sets and the distance from the set to the boundary of Ω tend
to zero when s →∞. In this domain the following nonlinear problem was
studied:
⎧
⎪⎨
⎪⎩
n∑
j=1
d
dx j
f j (x, u s (x), ∇u s (x)) − f 0 (x, u s (x), ∇u s (x)) =
= λ s g 0 (x, u s (x)), x ∈ Ω s ,
u s (x) = 0, x ∈ ∂Ω s .
(1.1)
It was proved that the following boundaryvalue problem is the limit for
3
(1.1) under some additional conditions on the functions g 0 ,f j ,j=0,...,n:
⎧ n∑
d
⎪⎨ dx j
f j (x, u(x), ∇u(x)) − f 0 (x, u(x), ∇u(x)) + c 0 (x, −u(x)) =
j=1
= λg 0 (x, u(x)), x ∈ Ω,
⎪⎩
u(x) = 0, x ∈ ∂Ω.
(1.2)
Moreover, the convergence of the eigenelements of the original problems to
the eigenelements of the corresponding limit problem was established: let
λ s be the eigenvalue of (1.1) and λ be the eigenvalue of (1.2) and let u s (x)
and ũ(x) be the corresponding eigenfunctions. Then lim λ s = λ and the
s→∞
sequence {u s (x)} ∞ s=1 converges when s →∞to ũ(x) strongly in W r 1 (Ω) for
any r
was considered in [9] and [22]. Here we suppose that the Dirichlet condition
holds on the boundary of cavities. In particular, we derive the limit
(homogenized) problem for the original problems. Moreover, we establish
the strong convergence in H 1 of the solutions for the considered problems
to the corresponding solution of the limit problem. Moreover, we prove that
the eigenelements of the spectral problems converge to the corresponding
eigenelements of the spectral limit problem. It should be noted that we use a
general scheme for the proof of the convergence, which was developed in [5],
[7], [12] and [46]. Moreover, we apply these results to obtain some independently
interesting facts concerning the constant in our version of Friedrichs
inequality.
The following estimate is known (see e.g. [18] and [38, Ch.III, §5]) as the
Friedrichs inequality for functions from the Sobolev space H ◦ 1 (Ω) :
∫
∫
u 2 dx ≤ K 0 ∇u 2 dx,
Ω
Ω
where the constant K 0 depends only on the domain Ω. A very closely related
result is the Poincaré inequality, which is valid for functions from H 1 (Ω):
∫
Ω
⎛
∫
u 2 dx ≤ K ⎝
Ω
∇u 2 dx + 1
Ω
⎛
∫
⎝
Ω
⎞2⎞
udx⎠
⎠ ,
where the constant K depends on the domain Ω.
Many mathematicians have been interested in these inequalities for various
reasons. Therefore a lot of different results on Friedrichstype and
Poincarétype inequalities have been obtained and applied in the literature.
In particular, in [60] a direct proof of Friedrichstype inequalities and
Poincarétype inequalities in some subspaces of Wp 1 (Ω) were given, where
Ω ⊂ R n is a bounded and connected open domain, n =2, 3. Some Poincaré
and Friedrichstype inequalities yielding for any v ∈ Wω 1,p (Ω) = {v ∈ Wp 1 (Ω) :
v ω =0}, 1 n.
5
We also note paper [39], where some properties of the best constant in
the following variational Friedrichs inequality
min ‖∇u‖ L p(Ω)
= λ pq (Ω) > 0inΩ,
‖u‖ Lq(Ω)
was studied, where the minimum is taken over all functions u ∈ Wp 1(Ω)
vanishing on the whole boundary ∂Ω or on some part of it.
In [36] the author established some general theorems concerning
Friedrichstype inequalities for functions from different spaces on an n
dimensional cube. A technique from the theory of isoperimetrical inequalities
was used to prove these theorems. Moreover, both upper and lower estimates
for the best constant were derived. It was also shown that the best constants
in such types of inequalities are closely related to the harmonic capacity (for
the definition see e.g. [36, Ch.I, §2]) of the set where the function vanishes.
The main result reads as follows: the Friedrichs inequality holds if and only if
the harmonic capacity is positive. This means that the Friedrichs inequality
is also valid (under some additional conditions concerning the geometry of
the domain) for functions from the Sobolev space H 1 (Ω), vanishing not on
the whole boundary of the domain but just on small pieces of it. In this
paper we give a direct proof of the Friedrichstype inequality for functions
from H 1 , vanishing on the boundary of the cavities in the periodic case and
show that the constant in this inequality is close to the one in the classical
Friedrichs inequality.
We also mentioned that some discrete analogues both of the Friedrichs
and Poincaré inequalities were proved in [54], [55] and [57]. It should be
noted that the results obtained in [57] can be applied to the analysis of nonconforming
numerical methods (for instance, for the discontinuous Galerkin
method).
The Friedrichstype inequality is a special case of the following multidimensional
Hardytype inequality:
⎛
⎞ 2
∫
⎝
R n
p(n−2)
−n+
x
2 u p dx⎠
p
∫
≤ C
R n
∇u(x) 2 dx,
where u(x) ∈ C0 ∞(Rn ),n>2, and 2 ≤ p ≤ 2n . n−2
This inequality was first generalized to domains in R n by J. Nečas in
[40]. He proved that if Ω is a bounded domain with Lipschitz boundary,
1
holds, where ϱ(x) =dist(x, ∂Ω). After that this inequality was generalized
by A. Kufner in [28] to domains with the Hölder boundary and later by
A. Wannebo (see [58]) to domains with the generalized Hölder condition.
After that a large number of complementary results and generalizations have
been proved. Indeed, Hardytype inequalities are today almost an art in its
own, see e.g. the books [29], [30] and [41] and the references given there.
In relation to this paper we just mention that some estimates for the best
constant in multidimensional Hardytype inequalities were proved in [1], [17]
and [23].
The classical Hardytype inequality can also be generalized to the following
form:
⎛
⎞ 1 ⎛
⎞ 1
∫
q ∫
p
⎝ V (x)u(x) q dx⎠
≤C⎝
W (x)∇u(x) p dx, ⎠ (1.3)
R n
R n
where u(x) ∈ C0 ∞(Rn ),V(x) ≥ 0,W(x) ≥ 0, p,q≥ 1, and the constant C
depends only on V (x) andW (x). There are several results also for this much
more general case (see e.g. the books mentioned above and the references
given there). Here we just mention that for the special case p = q =2it
was proved by V. Maz‘ya (see [36]) that in the case W (x) = 1 the following
condition is both necessary and sufficient for the validity of (1.3):
∫
V (x) dx ≤ βcap(F, Ω),
F
where cap(F, Ω) = inf{ ∫ ∇u(x) 2 dx}, F is a compact set and the infimum
Ω
is taken over the set
{u(x) ∈ C ∞ 0
(Ω), u ≡ 1 in a neighborhood of F, 0 ≤ u ≤ 1}.
For n = 1 the precise result reads as follows: The inequality
∫ ∞
∫ ∞
q(x)u 2 (x) dx ≤ c u ‘2 (x) dx, where u(0) = 0
0
0
is valid if and only if there exists a constant C 1 such that
sup x
x∈(0,∞)
∫ ∞
x
q(x) dx ≤ C 1 .
7
We pronounce that inequalities both of Hardytype and of Friedrichstype
are very important for many applications.
This paper is organized as follows: In Section 2 we present the geometrical
settings and state and discuss the main results. In Section 3 we prove the
Friedrichs inequality, which is stated in Theorem 2.1. An important Lemma
(Lemma 2.1) which is used in the proofs of some of the main theorems, is
proved in Section 4. Sections 5 and 6 are devoted to the proofs of Theorems
2.2 and 2.3, respectively. This means that we have investigated the convergence
of solutions as well as the convergence of the respective eigenelements
in a domain perforated periodically along the boundary. Some auxiliary estimates
for the solutions of such problems are derived in Section 7. Finally,
Section 8 we reserved for some concluding remarks.
2 The main results.
Let us denote by Ω the cube
{x =(x 1 ,x 2 ,x 3 ) ∈ R 3 :0 ≤ x 1 ≤ c, 0 ≤ x 2 ≤ c, 0 ≤ x 3 ≤ c}.
Assume that ε = 1 is a small parameter, N is a natural number, N≫1.
N
Further we use also the following notations.
Let B := {ξ : ξ1 2 + ξ2 2 +(ξ 3 − 1) 2
Figure 1: Domain perforated along the boundary.
Our aim is to derive the Friedrichs inequality for functions
u ε ∈ H 1 (Ω ε , Γ ε ) and to prove some new homogenization theorems. Our
main results read as follows:
Theorem 2.1. The inequality
∫
∫
u 2 ε dx ≤M
Ω ε
Ω ε
∇u ε  2 dx (2.1)
holds for any functions u ε
depend on ε.
∈ H 1 (Ω ε , Γ ε ), where the constant M does not
Theorem 2.2. Assume that F ∈ L 2 (Ω) and K is an arbitrary compact set
belonging to the complex plane C, Kdoes not contain the eigenvalues of the
problem
⎧
⎪⎨ −Δu 0 = λ 0 u 0 in Ω,
u 0 =0 on Γ 0 ,
(2.2)
⎪⎩ ∂u 0
=0 on Γ \ Γ ∂ν 0.
Then the following statements hold:
9
1. There exists a number ε 0 > 0, such that the unique solution to the
problem
⎧
⎪⎨ −ΔU ε = λU ε + F in Ω ε ,
U ε =0 on Γ ε ,
(2.3)
⎪⎩ ∂U ε
=0 on Γ, ∂ν
does exist for all ε
The proofs of these theorems are based on a method used in [5], [7], [19],
[20], [21] and [46] for different types of singular perturbations. However,
we must modify this method according to the geometrical structure of our
domain. The following Lemma, which is proved in Section 4, is necessary for
our analysis:
Lemma 2.1. Let v ε be a sequence of functions from H 1 (Ω ε , Γ ε ) and assume
that v ε ⇀v ∗ weakly in H 1 (Ω) when ε → 0. Then v ∗ ∈ H 1 (Ω, Γ 0 ).
Further, we denote by (X, Y) the inner product of the vectors X and
Y. We consider weak solutions to the problems (2.2), (2.3), (2.6) and (2.7)
(see e.g. [52]). The function U ε ∈ H 1 (Ω ε , Γ ε ) is a weak solution to the
boundaryvalue problem (2.3) if the integral identity
∫
∫ ∫
(∇U ε , ∇v) dx − λ U ε vdx= F vdx (2.8)
Ω ε
Ω ε
Ω ε
holds for any v ∈ H 1 (Ω ε , Γ ε ); the function U 0 ∈ H 1 (Ω, Γ 0 ) is a weak solution
to the boundaryvalue problem (2.6) if the integral identity
∫
∫ ∫
(∇U 0 , ∇v) dx − λ U 0 vdx= F vdx (2.9)
Ω
Ω
Ω
is valid for any v ∈ H 1 (Ω, Γ 0 ). Analogously, the function u ε ∈ H 1 (Ω ε , Γ ε )
is an eigenfunction to the spectral problem (2.7), corresponding to λ ε ,ifthe
integral identity ∫
∫
(∇u ε , ∇v) dx = λ ε u ε vdx
Ω ε
holds for any v ∈ H 1 (Ω ε , Γ ε ); the function u 0 ∈ H 1 (Ω, Γ 0 ) is an eigenfunction
to the spectral problem (2.2), corresponding to λ 0 , if the integral identity
∫
∫
(∇u 0 , ∇v) dx = λ 0 u 0 vdx
Ω ε
Ω
Ω
is valid for any v ∈ H 1 (Ω, Γ 0 ).
Note that, due to the Remark 2.1, the integral identity (2.8) can be
rewritten as follows:
∫
∫ ∫
(∇U ε , ∇v) dx − λ U ε vdx= F vdx. (2.10)
Ω
Ω
Ω
11
3 Proof of Theorem 2.1.
Proof. Let us represent the domain Ω ε in the following way:
Ω ε = ⋃ i,j
Π i,j
1ε ∪ Π i,j
2ε ,
where Π i,j
1ε is the parallelepiped of size 2aε × 2aε × c, containing the ball Bε i,j ,
and Π i,j
2ε is the parallelepiped of size (1 − 2a)ε × (1 − 2a)ε × c, having the
common vertical bounds with Π i,j
Define also Π ij,+
1ε
by the top half of the ball B i,j
ε
x 3 = c. Analogously, Π ij,−
1ε
by the lower half of the ball B i,j
ε
x 3 =0. Let us denote by Π ω,ij,+
1ε
where ω < c; Γ ω,ij
1ε
Γ ω,ij
1ε = {(x 1 ,x 2 ,x 3 ) ∈ Π ij,−
way we define the sets Π ω,ij
2ε
1ε , Πi+1,j 1ε
, Π i,j−1
1ε and Π i,j+1
1ε .
1ε, which is bounded from below
as the part of Π i,j
and bounded from above by the plane
is the part of Π i,j
1ε , which is bounded from above
and bounded from below by the plane
the domain {(x 1 ,x 2 ,x 3 ) ∈ Π ij,+
1ε ,x 3 ε, and the
12
1ε
espective point (x 1 ,x 2 ,x 0 3) ∈ Γ ε belonging to the top half of the ball B i,j
ε .
Using the NewtonLeibnitz formula, we get that
u ε (x 1 ,x 2 ,ω)=u ε (x 1 ,x 2 ,ω) − u ε (x 1 ,x 2 ,x 0 3)=
∫ ω
x 0 3
∂u ε
∂x 3
dx 3 ,
since u ε (x 1 ,x 2 ,x 0 3 )=0.
Moreover, according to the CauchySchwartzBunyakovsky inequality, we
have that
u 2 ε(x 1 ,x 2 ,ω)=
⎛
⎜
⎝
∫ ω
x 0 3
⎞
∂u ε ⎟
dx 3 ⎠
∂x 3
2
∫ ω
≤ ω
≤ ω
x 0 3
∫ ω
( ∂uε
∂x 3
) 2
dx 3 ≤
∇u ε  2 dx 3 .
(3.1)
x 0 3
Then, by integrating both parts of the inequality (3.1) with respect to x 1 ,x 2
over Γ ω,ij
1ε , we obtain that
∫
∫
u 2 ε (x 1,x 2 ,ω) dx 1 dx 2 ≤ ω ∇u ε  2 dx. (3.2)
Γ ω,ij
1ε
Π ω,ij,+
1ε
It follows from (3.2) that
∫
u 2 ε(x 1 ,x 2 ,ω) dx 1 dx 2 ≤ c
∫
∇u ε  2 dx. (3.3)
Γ ω,ij
1ε
Π ij,+
1ε
Finally, by integrating (3.3) with respect to ω over [ε, c], we get that
∫
∫
dx ≤ c2 ∇u ε  2 dx. (3.4)
u 2 ε
Π ij,+
1ε
Π ij,+
1ε
The case when (x 1 ,x 2 ,ω) ∈ Π ij,−
1ε ,ω < ε, can be considered analogously,
so we omit the details. In this case an analogous inequality can be obtained:
∫
∫
u 2 ε dx ≤ c 2 ∇u ε  2 dx. (3.5)
Π ij,−
1ε
Π ij,−
1ε
13
Consider now the parallelepiped Π i,j
2ε and assume that the ball Bε
i,j
touches it from the left. For Π 0,j
2ε we consider the ball Bε
1,j . Assume that
ω > ε (the case when ω < ε can be considered analogously). Connect
all points of Γ ω,ij
2ε to the bound of Bε
i,j . Then we obtain the beam of
lines with the directions l(x) (see Figure 3). Let (x 1 ,x 2 ,ω) ∈ Γ ω,ij
2ε and
Figure 3: The tangent lines.
(˜x 1 , ˜x 2 , ˜x 3 ) ∈ ∂B i,j
ε ∩ l(x 1 ,x 2 ,ω). From the geometrical constructions the
distance between the points (x 1 ,x 2 ,ω)and(˜x 1 , ˜x 2 , ˜x 3 )islessthenorequal
to √ (ω − ε) 2 +(1+4a 2 )ε 2 . Since u ε (˜x 1 , ˜x 2 , ˜x 3 ) = 0, by using the Newton
Leibnitz formula, we have that
u ε (x 1 ,x 2 ,ω)=
(x 1 ∫,x 2 ,ω)
(˜x 1 ,˜x 2 ,˜x 3 )
∂u ε
∂l
dl.
Then, by squaring both sides of this formula and using the CauchySchwartz
14
Bunyakovsky inequality, we obtain the following inequality:
u 2 ε (x 1,x 2 ,ω) ≤ √ (ω − ε) 2 +(1+4a 2 )ε 2
(x 1 ∫,x 2 ,ω)
(˜x 1 ,˜x 2 ,˜x 3 )
≤ C 1 ω
( ) 2 ∂uε
dl ≤
∂l
(x 1 ∫,x 2 ,ω)
(˜x 1 ,˜x 2 ,˜x 3 )
∇u ε  2 dl,
where the constant C 1 does not depend on ε.
Integrating both parts of this inequality with respect to x 1 ,x 2 over Γ ω,ij
2ε
and replacing the righthand side by the greater integral, we get that
∫
∫
u 2 ε (x 1,x 2 ,ω) dx 1 dx 2 ≤ C 1 ω
∇u ε  2 dx. (3.6)
Γ ω,ij
2ε
It follows from (3.6) that
∫
u 2 ε dx 1 dx 2 ≤ C 1 c
Γ ω,ij
2ε
Π ω,ij
2ε
∪Πω,ij,+ 1ε ∪Π ω,ij,−
1ε
∫
Π ij
2ε ∪Πij,+ 1ε
∪Πij,− 1ε
∇u ε  2 dx. (3.7)
Now, by integrating both parts of the inequality (3.7) with respect to ω
over [0,c], we find that
∫
∫
u 2 ε dx ≤ C 1c 2
∇u ε  2 dx. (3.8)
Π ij
2ε
Π ij
2ε ∪Πij,+ 1ε
∪Πij,− 1ε
Summarizing the inequalities (3.4), (3.5) and (3.8), we obtain that
∫
∫
u 2 ε dx ≤C 1c 2
Ω ε
Ω ε
∇u ε  2 dx. (3.9)
Approximating the functions from H 1 (Ω ε , Γ ε ) by smooth functions, we
conclude that inequality (3.9) is valid for u ε ∈ H 1 (Ω ε , Γ ε ). The inequality
(2.1) follows directly from (3.9). The proof is complete.
4 Proof of Lemma 2.1.
Proof. Let us define the following sets:
Γ ω =Ω ε
⋂
{(x1 ,x 2 ,x 3 ):x 3 = ω}, i.e. Γ ω = ⋃ i,j
(
Γ
ω,ij
1ε
)
∪ Γ ω,ij
2ε
15
and
Π ω =Ω ε
⋂
{(x1 ,x 2 ,x 3 ):x 3
Proof. Indeed, if ‖U ε ‖ 0 ≠1, then by setting V ε =
and the function V ε satisfies the problem
{
−ΔV ε = λV ε + F ε in Ω,
V ε =0 onΓ 4 ,
Uε
‖U ε‖ 0
, we obtain that
‖V ε ‖ 0 =1 (5.2)
where
F ε =
F . (5.3)
‖U ε ‖ 0
Hence, due to the assumptions we see that the estimate
‖V ε ‖ 1 ≤ C‖F ε ‖ 0
holds for V ε . Multiplying the last inequality by ‖U ε ‖ 0 and using (5.2) and
(5.3), we obtain the estimate (5.1) for any U ε . The proof is complete.
Proof of Theorem 2.2. First, we prove the following estimate:
‖U ε ‖ 1 ≤ C 7 (‖U ε ‖ 0 + ‖F ‖ 0 ), (5.4)
where C 7 does not depend on ε and λ. Substituting v = U ε in the integral
identity (2.10), we obtain that
‖U ε ‖ 2 1 ≤ C 8 (‖U ε ‖ 2 0 + ‖U ε ‖ 0 ‖F ‖ 0 ) ≤ C 7 (‖F ‖ 0 + ‖U ε ‖ 0 )‖U ε ‖ 1 ,
where C 7 does not depend on ε and λ. It is obvious that (5.4) follows from
this inequality.
1. The existence and uniqueness of the solution to Problem (2.3) follows
from the estimate (2.4).
Indeed, the integral identity (2.10) can be rewritten as follows :
∫
∫
∫ ∫
(∇U ε , ∇v) dx + U ε vdx− (λ +1) U ε vdx= F vdx. (5.5)
Ω
Ω
This means that the identity (5.5) is equivalent to the identity
(U ε ,v) 1 − (λ +1)(U ε ,v) 0 =(F, v) 0 . (5.6)
The Riesz theorem (see e.g. [59]) says that there exists an operator A, such
that (U ε ,v) 0 =(AU ε ,v) 1 . Hence, the identity (5.6) is equivalent to
(U ε ,v) 1 − (λ +1)(AU ε ,v) 1 =(F, v) 0 .
17
Ω
Ω
The solvability of this equation follows from the fact that the kernel of
the operator [I − (λ +1)A] is trivial (because the solution to the problem
(2.3) with zero righthand side is unique). The uniqueness of the solution to
(2.3) follows directly from (2.4).
Let us derive now the estimate (2.4). Assume that the estimate (2.4)
does not hold, i.e. that there exists a sequence {ε k } ε k → 0whenk →∞,
F k ∈ L 2 (Ω) and λ k such that the inequality
‖U εk ‖ 1 >k‖F k ‖ 0 (5.7)
holds for the solutions to the problem (2.3), where ε = ε k ,λ = λ k and
F = F k .
Due to the Lemma 5.1 we may assume without loss of generality that the
sequence {U ε } is normalized in L 2 (Ω), i.e. that
Then, by using (5.4) and (5.7), we obtain that
‖U εk ‖ 0 =1. (5.8)
‖U εk ‖ 1 ≤ C 11 ,
‖F k ‖ 0 < C 11
k , (5.9)
when k →∞.
It is known that a bounded set in H 1 is weakly compact in H 1 and, hence,
there exists a subsequence {k ′ } of indexes {k} and U ∗ and λ ∗ such that
λ k ′ → λ ∗ ∈ K,
U εk ′
⇀U ∗ in H 1 (Ω) when k ′ → +∞. (5.10)
The inequalities (5.8) and (5.10) give that
U ∗ ≠0.
Suppose that v is an arbitrary fixed function from C0 ∞ (Ω, Γ 0 ). Then
v ∈ H 1 (Ω ε , Γ ε ) for all small ε. Substitute in the integral identity (2.10) v as
a test function, U = U εk ′
,λ= λ k ′,F = F k ′, when ε = ε k ′.Wehavethat
∫
(
∇Uεk
′
, ∇v ) ∫
∫
dx − λ k ′ U εk ′vdx= F k ′vdx. (5.11)
Ω
Ω
Ω
Passing to the limit in the integral identity (5.11) when ε k ′
(5.9) and (5.10), we obtain the following integral identity:
∫
∫
(∇U ∗ , ∇v) dx = λ ∗ U ∗ vdx.
→ 0 and using
Ω
Ω
18
From the density of the embedding C0 ∞ (Ω, Γ 0 )intoH 1 (Ω, Γ 0 )wecan
conclude that this inequality holds also for v ∈ H 1 (Ω, Γ 4 ). It follows from
Lemma 2.1 that U ∗ ∈ H 1 (Ω, Γ 0 ). Due to the fact that U ∗ ≠0andv is an
arbitrary function from H 1 (Ω, Γ 0 ), it follows that λ ∗ ∈ K is the eigenvalue
of the limit problem (2.2). But we assumed that K did not contain the
eigenvalues of the limit problem (2.2). This contradiction proves the estimate
(2.4).
2. Let λ ∈ K be an arbitrary fixed number and assume that the sequence
{ε k }→0whenk → +∞. A subsequence, which is weakly convergent in H 1 ,
can always be chosen from a sequence which is bounded in H 1 . Hence, there
is a subsequence {k ′ } of indexes {k} and a function U ∗ such that U εk ′
⇀U ∗
in H 1 (Ω) (and strongly in L 2 (Ω)) when k ′ → +∞.
By now using Lemma 2.1 we obtain that the limit function
U ∗ ∈ H 1 (Ω, Γ 0 ). Passing to the limit in the integral identity (5.11) by
means of the same reasoning as in the proof of Part 1 of the Theorem, we
get that ∫
∫ ∫
(∇U ∗ , ∇v) dx = λ ∗ U ∗ vdx+ F vdx,
Ω
which coincides with the integral identity of the problem (2.6). Since the
solution to the problem (2.6) is unique we conclude that U ∗ = U 0 , and,
moreover, from (5.10) we find that
Ω
U ε → U 0 strongly in L 2 (Ω) and weakly in H 1 (Ω) when ε → 0. (5.12)
Substituting v = U εk in the integral identity (5.5) and using the convergence
of the function in L 2 (Ω), we obtain that
∫
∫
‖∇U ε ‖ 2 0 −‖∇U 0‖ 2 0 = F U ε dx − F U 0 dx =
Ω
Ω
∫
=
Ω
Ω
F (U ε − U 0 ) dx −→ 0 when ε → 0.
(5.13)
The convergence (2.5) follows from (5.12) and (5.13). Indeed, by using
(5.12) and (5.13), we have that
∫
‖∇(U ε − U 0 )‖ 2 0 =‖∇U ε ‖ 2 0 + ‖∇U 0 ‖ 2 0 − 2Re (∇U ε , ∇U 0 )dx → 0
when ε → 0. The convergence (2.5) follows from this convergence and from
(5.12). The proof is complete.
Ω
19
6 Proof of Theorem 2.3.
The following Lemma can be proved analogously to the proof of statement
2) of Theorem 2.2:
Lemma 6.1. Assume that λ ε is the eigenvalue of the problem (2.7), converging
to the eigenvalue λ 0 of the limit problem (2.2) when ε → 0, and
let u ε be the eigenfunction, which is normalized in L 2 (Ω) and corresponding
to the eigenvalue λ ε . Then the sequence {ε k } ∞ k=1 → 0 has a subsequence
{ε km } ∞ m=1 → 0 such that ‖u εkm − u 0 ‖ 1 → 0,
where u 0 is the eigenfunction, which is normalized in L 2 (Ω) and corresponding
to the eigenvalue λ 0 .
The mutual orthogonality of the eigenfunctions of the problem (2.2) as
well as of the problem (2.7) and Lemma 6.1 give the following statement:
Lemma 6.2. Let the multiplicity of the eigenvalue λ 0 to the problem (2.2) be
equal to N and suppose that the eigenvalues λ ε of the problem (2.7) converge
to λ 0 . Then the multiplicity of λ ε is less then or equal to N.
Proof of Theorem 2.3. Let
0
According to Lemmas 6.1 and 6.2 it is enough to prove that N ε ≥ N,
where N ε is the multiplicity of the eigenvalues λ i ε converging to λ 0 when
ε → 0andN is the multiplicity of the eigenvalue λ 0 .
Suppose the contrary. This means that there exists a sequence
{ε k } ∞ k=1 → 0 such that N ε k
0, such that S(t, λ 0 ) does not contain the eigenvalues
of the limit problem except λ 0 as t
The representation (6.4) leads us to the following:
G 0 (λ) = 1
λ 0 − λ ,
M∑
G ε (λ) =
i=1
(u p+M+1
0 ,u p+i
ε ) 0  2
λ p+i
ε
− λ
+ g ε (λ),
(6.5)
where
g ε (λ) =
p∑
s=1
(u p+M+1
0 ,u s ε) 0  2
λ s ε − λ +
∞∑
s=p+M+1
(u p+M+1
0 ,u s ε) 0  2
λ s ε − λ . (6.6)
Let us show that the function g εkm (λ) is holomorphic as λ ∈ S(t, λ 0 )for
any t
since t0. Hence, the function U δ can be represented in a
Fourier series of the orthogonal basis (7.1) in the following form:
U δ =
∞∑
s=1
(U δ ,u s δ ) 1,δ
u s
λ s δ = ∑ ∞
δ
s=1
23
(U δ ,u s δ ) 1,0
u s
λ s δ , δ ≥ 0, (7.2)
δ
in H 1 (Ω, Γ 0 ) for any function U 0 ∈ H 1 (Ω, Γ 0 )whenδ =0andinH 1 (Ω ε , Γ ε )
for any function U ε ∈ H 1 (Ω ε , Γ ε )whenδ = ε>0. The integral identities
(2.9) and (2.10) lead to the following formula:
(U δ ,u s δ ) 1,0 = λ(U δ ,u s δ ) 0 +(F, u s δ ) 1,0.
Substituting the representation (7.2) in the righthand side of this equation,
we obtain that
(U δ ,u s δ ) 1,0 = λ (U
λ s δ ,u s δ ) 1,0 +(F, u s δ ) 0.
0
Hence,
(U δ ,u s δ) 1,0 = λs 0
λ s 0 − λ (F, us δ) 0 .
Substituting this formula in (7.2), we obtain the representations (6.4).
Proposition 7.1. Let λ 0 be the eigenvalue of the Dirichlet problem (2.2) of
multiplicity N:
λ 0 = λ p+1
0 = ...= λ p+N
0
holds and is uniform in ε and λ. It follows from (6.4) that
ũ ε =
p∑
s=1
(F, u s ε) 0
λ s ε − λ us ε +
∞
∑
s=p+N+1
(F, u s ε) 0
λ s ε − λ us ε . (7.4)
The estimate (7.4) and the orthogonality of the system of functions u s ε,
s = 1, 2,... in L 2 (Ω) lead to the fact that the function ũ ε is orthogonal in
L 2 (Ω) to uε
p+i , i =1,...,N.
Let us derive the estimate (7.3) for ũ ε . Using the orthogonality of the
system (7.1) in H 1 (Ω) with respect to the inner product (•, •) 1,0 and using
the representation (7.4), we have that
p∑
‖ũ ε ‖ 2 1,0 =
(F, u s ε ) ∣
0 ∣∣∣
2
∞∑
∣ λ s s=1 ε − λ ‖u s ε‖ 2 1,0 +
(F, u s ε ) ∣
0 ∣∣∣
2
∣ λ s s=p+N+1 ε − λ ‖u s ε‖ 2 1,0
p∑
= ‖ũ ε ‖ 2 1,0 =
(F, u s ε ) ∣
0 ∣∣∣
2
∞∑
∣ λ s ε − λ λ s ε +
(F, u s ε ) ∣ (7.5)
0 ∣∣∣
2
∣ λ s ε − λ λ s ε.
s=1
s=p+N+1
It is evident that there exists a number σ>0 such that the inequality
λ s ε
− λ >σ, s≠ p +1,...,p+ N +1, (7.6)
holds for all small ε when λ is close to λ 0 .
Due to (7.6) and the ParsevalSteklov formula, we can derive from (7.5)
that
‖ũ ε ‖ 2 1,0 ≤ ∑ ∞ σ−1 (F, u s ε ) 0 2 ≤ C 1 ‖F ‖ 2 0 .
s=1
Using this inequality and the fact that the norms ‖ũ ε ‖ 1,0 and ‖ũ ε ‖ 1 are equivalent,
we obtain (7.3). The proof is complete.
8 Concluding remarks.
Remark 8.1. Using Theorems 2.2 and 2.3, we conclude that the
following Friedrichs inequalities with precise constants hold true for
u ε ∈ H 1 (Ω ε , Γ ε ),u 0 ∈ H 1 (Ω, Γ 0 ):
∫
∫
u 2 ε dx dy ≤M ε ∇u ε  2 dx dy
Ω ε
∫
u 2 0 dx dy ≤M 0
Ω ε
∫
∇u 0  2 dx dy,
Ω
Ω
25
where the constant M 0 does not depend on ε and
M ε −M 0 →0asε → 0.
This follows from the fact that M ε = 1 and M
λ 1 0 = 1 , which can be derived
ε
λ 1 0
from the variational principle for the problems (2.7) and (2.2), respectively.
Remark 8.2. Similar results as we proved in this and the previous sections
can be obtained also in the case when Ω ε is an arbitrary three dimensional
domain perforated periodically along the boundary.
Acknowledgments.
I thank Professor LarsErik Persson for his valuable remarks and help in
preparation of this publication. It is also a pleasure for me to thank Professor
Gregory A. Chechkin for setting of the problems and for his attention to my
work on this paper.
The work was partially supported by RFBR (project 090100353) and
by the program “Leading Scientific Schools” (HIII1698.2008.1)and by Luleå
University of Technology (Sweden).
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Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
&
Department of Mathematics
Luleå University of Technology
SE971 87 Luleå, Sweden
korolevajula@mail.ru
Paper D
On the Eigenvalue in a Domain Perforated
Along the Boundary
G.A.Chechkin, R.R.Gadyl’shin and Yu..Koroleva
Presented by Academician A.M.Il’in December 2, 2009
1 Introduction
The homogenization of problems for elliptic operators in domains with small
orifices has been addressed in fairly numerous works (see, e.g., [1] — [10]
and the references therein). The basic goal is to determine a homogenized
(limiting) problem, compute the convergence rate of solutions to the perturbed
problem, and construct asymptotics of solutions (whenever possible).
The direction in which the eigenvalue of the perturbed problem shifts from
that of the homogenized problem can frequently be determined from variational
considerations. For example, in the case of contracting orifice with a
Dirichlet boundary conditions specified on it, the eigenvalue of the perturbed
problem is obviously large then that of the homogenized one. However, if
Neumann boundary conditions are not applicable. The asymptotic expansion
constructed for the eigenvalue of the perturbed problem in [11] has shown
that it can be different sides of the eigenvalue of the limiting problem (depending
on the chosen center of contraction of the orifice).
In this paper, we consider a singularly perturbed eigenvalue problem for
the operator −Δ in a twodimensional domain that is periodically perforated
along its boundary in the case when the diameter of the cavities, the distance
between them, and the distance to the boundary are of the same order of
smallness. A Dirichlet boundary condition is set on the boundaries of the
cavities, while the Neumann boundary condition is specified on the remaining
part of the boundary. For this problem, it can be shown that the homogenized
problem is one in a domain without orifices with a Dirichlet boundary
condition on that part of the boundary near which the small orifices were
situated (a detailed proof of this convergence can be found in [12]). Concerning
the displacement direction of the eigenvalue in this problem, variational
1
considerations are also ineffective, since the Dirichlet boundary conditions
on the orifices are replaced in the limit by boundary conditions of the same
type on part of the boundary.
For the perturbed problem, we construct a twoterm asymptotic expansion
of the eigenvalue that implies that the latter is less that the corresponding
eigenvalue of the homogenized problem.
2 Statement of the problem and the main result.
Consider Ω := {(x 1 ,x 2 ) ∈ R 2 , x 2 > 0}, ∂Ω := Γ, and Γ = Γ 1 ∪ Γ 2 ,
where Γ 1 := {(x 1 ,x 2 ):− 1 ≤ x 2 1 ≤ 1,x 2 2 =0} and Γ 2 is a smooth curve
coinciding with the lines x 1 = − 1 and x 2 1 = 1 near the ends of Γ 2 1, respectively.
Let B be an arbitrary twodimensional domain with a smooth
boundary that is symmetric about the vertical axis and lies in a disk K of
radius a< 1 centered at the point ( 0, 1 2 2)
. We introduce the small parameter
ε = 1 , where N ≫ 1, and N ∈ N. Define 2N +1 Bj ε := {x ∈ Ω :
ε −1 (x 1 − εj, x 2 ) ∈ B}, j ∈ Z, B ε = ⋃ Bε, j Γ ε = ∂B ε . Define the domain
j
Ω ε := Ω \ B ε (see figure 1).
Figure 1: The structure of the domain Ω ε and the periodicity cell Π.
Consider the following eigenvalue problems:
2
⎧
⎪⎨ −Δu ε = λ ε u ε in Ω ε ,
u ε =0 onΓ ε ,
(2.1)
⎪⎩ ∂u ε
=0 onΓ. ∂ν
⎧
⎪⎨ −Δu 0 = λ 0 u 0 in Ω,
u 0 =0 onΓ 1 ,
(2.2)
⎪⎩ ∂u 0
=0 onΓ\ Γ ∂ν 1.
Here and below, ν is an outward normal vector.
It can be shown that, if λ 0 is a simple eigenvalue of homogenized problem
(2.2), then the unique simple eigenvalue λ ε of perturbed problem (2.1) converges
to it as ε → 0 ( a detailed proof of convergence for problem (2.1) can
be found in [12]). Our goal is to construct a twoterm asymptotic expansion
of λ ε as ε → 0 that implies the inequality
λ ε 0.
The main result of this paper is the proof of the following theorem.
3
Theorem 2.1. Let λ 0 be a simple eigenvalue of problem (2.2). Then the
asymptotics of the eigenvalue λ ε converging to λ 0 as ε → 0 has the form
λ ε = λ 0 + ελ 1 + o(ε 3 2 −μ ) (2.6)
for any μ>0, where
∫
λ 1 = −C(B)
Γ 1
( ) 2 ∂u0
ds < 0. (2.7)
∂ν
Note that (2.6) and (2.7) imply inequality (2.3).
Asymptotics will be constructed by the method of matched asymptotic
expansions [5]. Throughout the rest of this paper, the eigenfunction u 0 is
assumed to be normalized in L 2 (Ω). Since (2.2) is a limiting problem for
(2.1), the asymptotics of the eigenvalue λ ε and the eigenfunction u ε outside
a small neighborhood of Γ 1 (outer expansion) are naturally sought in the form
λ ε ≈ ̂λ ε = λ 0 + ελ 1 + ε 2 λ 2 (2.8)
u ε (x) ≈ û ε = u 0 (x)+εu 1 (x)+ε 2 u 2 (x) (2.9)
Clearly, u 0 ∈ C ∞ (Ω), and problem (2.2) implies the equality
u 0 (x) =α 1 0 (x 1)x 2 + O(x 3 2 ) as x 2 → 0, (2.10)
where
∣
α0 1 = ∂u ∣∣∣∣x2 [
0
∈ C ∞ − 1 ∂x 2 2 , 1 ]
,
2
=0
d 2j+1 α 1 0
dx 2j+1
1
∣
∣
x1 =± 1 2
=0. (2.11)
The coefficients u 1 (x) andu 2 (x) are sought by solving the boundaryvalue
problems ⎧⎪ ⎨
⎪ ⎩
−Δu 1 = λ 0 u 1 + λ 1 u 0 in Ω,
∂u 1
∂ν =0 onΓ 2,
u 1 = α1 0 on Γ 1 ,
⎧
−Δu ⎪⎨ 2 = λ 0 u 2 + λ 1 u 1 + λ 2 u 0 in Ω,
∂u 2
⎪⎩
∂ν =0 onΓ 2,
u 2 = α2 0 on Γ 1 ,
where α 0 1 (x 1)andα 0 2 (x 1) are as yet unknown (arbitrary) functions.
(2.12)
(2.13)
4
The equations in problems (2.12), (2.13) and the boundary conditions
outside Γ 1 are obtained by substituting sums (2.9) and (2.8) into problem
(2.1) and formally equating the coefficients of like powers of ε up to the
second order inclusive. Outside a small neighborhood of Γ 1 , we use outer
expansion (2.9). For this reason, the boundary conditions on Γ 1 are as yet
arbitrary Dirichlet conditions.
In what follows, α1 0 and α0 2 are assumed to satisfy the conditions
[
αi 0 ∈ C∞ − 1 2 , 1 ]
(
, (αi 0 2
)(2j+1) ± 1 )
=0, j =0, 1,.... (2.14)
2
Then there exist constants λ 1 and λ 2 and functions u 1 (x),u 2 (x) ∈ C ∞ (Ω)
that solve problems (2.12) and (2.13); moreover,
λ 1 = −
1
∫2
α 0 1(x 1 )α 1 0(x 1 ) dx 1 . (2.15)
Therefore, first,
− 1 2
u 1 (x) =α1 0 (x 1)+α1 1 (x 1)x 2 + O(x 2 2 ),u 2(x) = α2 0 (x 1)+O(x 2 ),x 2 → 0,
(2.16)
where
[
α1 1 ∈ C∞ − 1 2 , 1 ]
(
, (α1 1 2
)(2j+1) ± 1 )
=0, (2.17)
2
and, second, û ε belongs to C ∞ (Ω) and satisfies
⎧
⎨ −Δû ε = ̂λ ε û ε + O(ε 3 ) in Ω,
∂û
⎩ ε
=0 onΓ 2 .
∂ν
(2.18)
Now we construct an inner expansion in a small neighborhood of Γ 1 and
simultaneously determine the boundary conditions αj 0 . Putting (2.10) and
(2.16) into (2.9) and making the substitution ξ 2 = x 2
ε
yields
û ε (x) =εV 1 (ξ 2 ; x 1 )+ε 2 V 2 (ξ 2 ; x 1 )+O(x 3 2 + εx2 2 + ε2 x 2 ), (2.19)
as εξ 2 = x 2 → 0, where
V 1 (ξ 2 ; x 1 )=α 1 0 (x 1)ξ 2 + α 0 1 (x 1), V 2 (ξ 2 ; x 1 ) = α 1 1 (x 1)ξ 2 + α 0 2 (x 1). (2.20)
Following the method of matched asymptotic expansions, we conclude that
the inner expansion must have the structure
u ε (x) ≈ ̂v ε (x) =εv 1 (ξ; x 1 )+ε 2 v 2 (ξ; x 1 ) , (2.21)
5
where ξ = x ε , and v q (ξ; x 1 ) ∼ V q (ξ 2 ; x 1 ) as ξ 2 → +∞. (2.22)
Next, the terms of inner expansion (2.21) are constructed as 1periodic
functions of ξ 1 (for each fixed value of x 1 ).
Rewriting the equation and the boundary conditions to problem (2.1) in
coordinates ξ and substituting inner expansion (2.21) and sum (2.8) into the
equation, we combine like powers of ε (specifically, ε q−1 in the equation and
ε q in the boundary conditions, where q 1). Taking into account (2.22) and
(2.20) gives the following problems for v 1 and v 2 :
⎧
Δ ξ v 1 =0 inΠ\B,
⎪⎨ v 1 =0 on∂B,
∂v 1
∂ξ 2
=0 onγ,
(2.23)
∂v 1
∂ξ 1
(ξ; ± 1 2
⎪⎩
)=0 asξ 1 = ± 1,
2
v 1 ∼ V 1 as ξ 2 → +∞
⎧
−Δ ξ v 2 =2 ∂2 v 1
∂x 1 ∂ξ 1
in Π\B,
⎪⎨ v 2 =0 on∂B,
∂v 2
∂ξ 2
=0 onγ,
∂v 2
∂ξ 1
(ξ; ± 1 2
⎪⎩
)=− ∂v 1
∂x 1
(ξ; ± 1) as ξ 2 1 = ± 1,
2
v 2 ∼ V 2 as ξ 2 → +∞.
(2.24)
We analyze the solvability of problems (2.23) and (2.24) and simultaneously
determine the functions α 0 1 (x 1)andα 0 2 (x 1). Define
X(ξ) =Y (ξ)+ξ 2 .
The definition of Y implies that X can be extended as a 1periodic function
of ξ 1 . Retaining the same notation for the extended function, we set
v 1 (ξ; x 1 )=α 1 0 (x 1)X(ξ). (2.25)
Then, by virtue of (2.4) and (2.20), v 1 is a 1periodic function of ξ 1 that
solves problem (2.23) and
v 1 (ξ; x 1 )=V 1 (ξ 2 ; x 1 )+O(e −2πξ 2
), as ξ 2 → +∞, (2.26)
for α 0 1 (x 1)=α 1 0 (x 1)C(B). (2.27)
6
Note that, in view of (2.11), (2.27) and (2.25), equalities (2.14) hold for α1
0
and
∂v 1
(ξ; x 1 )=0atx 1 = ± 1 ∂x 1 2 . (2.28)
Therefore, first, by virtue of (2.15), (2.11) and (2.27), we have formula (2.7)
and, second, equality (2.17) holds true.
Consider problem (2.23). Following [13], it is easy to show that the
boundaryvalue problem
⎧
Δ ξ ˜X =
∂X
∂ξ 1
in Π\B,
⎪⎨ ˜X =0 on∂B,
∂ ˜X
(2.29)
∂ξ 2
=0 onγ,
⎪⎩ ˜X =0 asξ 1 = ± 1 2
has a solution that is odd in ξ 1 with the asymptotics
˜X(ξ) =O(e −ξ 2
)asξ 2 →∞. (2.30)
Since the righthand side of the equation and the boundary conditions
in (2.29) are 1periodic, ˜X can be extended as a 1periodic function of ξ1 .
Retaining the same notation for the extended function, we set
v 2 (ξ; x 1 )=α 1 1(x 1 )X(ξ) − 2(α 1 0) ′ (x 1 ) ˜X(ξ). (2.31)
Then, by virtue of problems (2.29) and (2.4) and equalities (2.5), (2.25),
(2.28), (2.30) and (2.20), we conclude that v 2 is a 1periodic function of ξ 1
that solves problem (2.24) with the asymptotics
v 2 (ξ; x 1 )=V 2 (ξ 2 ; x 1 )+O(e −ξ 2
)asξ 2 →∞, (2.32)
for
α 0 2(x 1 )=α 1 1(x 1 )C(B).
It follows from (2.17) and (2.32) that condition (2.14) also holds for α 0 2 .
The formal construction of asymptotically matched approximations is
completed. Now we substantiate the resulting approximations. In what
follows, 0
where
‖ ̂F
)
ε v ‖ L 2 (Ω ε∩{x 2
References
[1] V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains
with FineGrained Boundaries (Naukova Dumka, Kiev, 1974) [in
Russian].
[2] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for
Periodic Structures (NorthHolland, Amsterdam, 1978).
[3] N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes
in Periodic Media (Nauka, Moscow, 1984; Kluwer, Dorderecht,
1989).
[4] E. SánchezPalencia, Homogenization Techniques for Composite Media
(SpringerVerlag, Berlin, 1987).
[5] A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary
Value Problems (Nauka, Moscow, 1989; Am.Math.Soc., RI, Providence,
1992).
[6] O.A.Oleinik, A.S.Shamaev and G.A.Yosifian Mathematical Problems
in Elasticity and Homogenization (Mosk. Gos. Univ., Moscow, 1990;
NorthHolland, Amsterdam, 1992).
[7] D. Cioranescu and P. Donato, An Introduction to Homogenization (Oxford
University Press, Oxford, 1999).
[8] V. Maz’ya, S. Nazarov and B. Plamenevsij, Asymptotic Theory of
Elleptic Boundary Value problems in Singularly Perurbed Domains(
Birkäuser, Basel, 2000), Vol.1.
[9] V. A. Marchenko and E. Ya. Khruslov, Homogenized Models of Microinhomogeneous
Media (Naukova Dumka, Kiev, 2005) [in Russian].
[10] A. L. Pyatnitskii, A. S. Shamaev and G. A. Chechkin, Homogenization:
Methods and Applications (Tamara Rozhkovskaya, Novosibirsk, 2007)
[in Russian].
[11] V. Maz’ya, S. Nazarov and B. Plamenevsij, Asymptotic expansions of
eigenvalues of boundary value problems for the Laplace operator in
domains with small openings, Izv. Acad. Nauk SSSR, Ser. Mat. 48,
347–371 (1984).
9
[12] R. R. Gadyl’shin, G. A. Chechkin and Yu. . Koroleva, On the Convergance
of the Solutions and Eigenelements of the Boundary–Value Problem
in the Domain Perforated Along the Boundary, Differ. Uravn. 46,
5–19 (2010).
[13] S. A. Nazarov, Junctions of singularly degenerating domains with different
limit dimensions, 1, Tr. Semin. im.I.G.Petrovskogo 18, 3–78 (1995).
[14] T. Kato, Perturbation Theory for Linear Operators (SpringerVerlag,
Berlin, 1966; Mir, Moscow, 1972).
Paper E
On the Weighted Hardy Type Inequality in a
Fixed Domain for Functions Vanishing on the
Part of the Boundary.
Yu.O. Koroleva
Key words and phrases. Inequalities, Partial Differential Equations, Functional
Analysis, Spectral Theory, Homogenization Theory, Hardytype Inequalities.
AMS 2010 Subject Classification. 39A10, 39A11, 39A70,
39B62, 41A44, 45A05
Abstract
We derive and discuss a new twodimensional weighted Hardytype
inequality in a rectangle for the class of functions from the Sobolev
space H 1 vanishing on small alternating pieces of the boundary.
1 Introduction
Inequalities of Hardytype are very important for many applications. These
inequalities are important tools e.g. for deriving some estimates for operator
norms, for proving some embedding theorems, for estimating eigenvalues,
etc.
The following basic onedimensional Hardytype inequality is well known:
∫ a
0
( u
t
) ( ) p p ∫ p a
dt ≤ (u ′ ) p dt, (1.1)
p − 1 0
where u ∈ L p (0,a), u ′ ∈ L p (0,a), p>1, u(0) = 0.
This inequality could be generalized to the multidimensional weighted
form:
⎛
⎞ 1 ⎛
⎞ 1
∫
q ∫
p
⎝ V (x)u(x) q dx⎠
≤C⎝
W (x)∇u(x) p dx⎠
, (1.2)
R n
where n ∈ Z + , u(x) ∈ C ∞ 0 (Rn ),V(x) ≥ 0,W(x) ≥ 0, p,q ≥ 1, and
the constant C depends only on V (x) andW (x). There are several results
concerning weighted Hardytype inequality (see e.g. the books [10], [11] and
[14] and the references given there).
R n
1
For the case n =1, 1
2 Preliminaries.
Let Ω ⊂ R 2 be the rectangle [0,a] × [0,b]:
Ω={(x 1 ,x 2 ):0≤ x 1 ≤ a, 0 ≤ x 2 ≤ b}.
Assume that ε>0 is a small positive parameter, ε = b , N ≫ 1, N ∈ N,
N
and δ = const, 0
and the average value of the function u over B(·,r) ∈ R 2 is defined as
u B := 1 ∫
u(x) dx.
πr 2
Let u be a locally integrable function on R 2 . The maximal functions M(u)
and M R (u) ofu are defined by
M(u)(x) :=supu B ,M R (u)(x) := sup u B .
r>0
0
Proof. Fix the point (x 1 ,x 2 ) ∈ Π i 1. By using the NewtonLeibnitz formula,
we have
Hence,
u ε (x 1 ,x 2 )=u ε (x 1 ,x 2 ) − u(0,x 2 )=
⎛
u 2 ε (x 1,x 2 ) ≤ ⎝
∫ a
0
∫ x 1
⎞2
∫
∂u a
ε
dx 1
⎠ ≤ a
∂x 1
0
∂u ε
∂x 1
dx 1 ≤
0
∫ a
0
∇u ε  2 dx 1 .
∂u ε
∂x 1
dx 1 .
Then, by integrating the last inequality with respect to x 2 and after that
with respect to x 1 over Π i 1 , we obtain that
∫
∫
u 2 ε dx ≤ a2 ∇u ε  2 dx. (2.2)
Π i 1
Π i 1
Now choose the point (x 1 , ˜x 2 ) ∈ Π i 2 such that ˜x 2 = 1−δx
δ 2 + εδ. This means
that (x 1 , ˜x 2 ) ∈ Π i 2 if and only if (x 1,x 2 ) ∈ Π i 1 . By again using the Newton
Leibnitz formula, we fined that
Consequently,
u ε (x 1 , ˜x 2 )=u(x 1 ,x 2 )+
∫˜x 2
x 2
∂u ε
∂x 2
d˜x 2 .
∫˜x 2
u 2 ε(x 1 , ˜x 2 ) ≤ 2u 2 (x 1 ,x 2 )+2(˜x 2 − x 2 ) ∇u ε  2 d˜x 2 .
x 2
Integrating the last inequality over Π i 2 and substituting the integral on the
righthand side by the greater integral, we get that
∫
u 2 ε dx 1 d˜x 2 ≤ 2 1 − δ
δ
∫
u 2 ε dx 1 dx 2 +2ε 2 (1 − δ) 2 ∫
∇u ε  2 dx 1 d˜x 2 .
Π i 2
Π i 1
Π i 2
Finally, by applying the estimate (2.2) to the first integral on the righthand
side and substituting both integrals by the greater integral, we obtain that
∫ (
u 2 ε dx ≤ 2 a 2 1 − δ
) ∫
+ ε 2 (1 − δ) 2 ∇u ε  2 dx. (2.3)
δ
Π i 2
Π i 1 ∪Πi 2
5
By summarizing up the inequalities (2.2) and (2.3) with respect to i, we
obtain the desired estimate:
∫ ∫
(
u 2 ε dx = u 2 ε dx ≤ 2 a 2 1 − δ
) ∫
+ ε 2 (1 − δ) 2 ∇u ε  2 dx =
δ
Π 2
⋃
Π i 2
i
(
=2 a 2 1 − δ
δ
⋃
(Π i 1 ∪Πi 2)
i
+ ε 2 (1 − δ) 2 ) ∫
Ω
∇u ε  2 dx.
We also need the following wellknown Lemmas:
Lemma 2.2. Let u ∈ W1 1 (B). Then
∫
u(x) − u B ≤2
B
∇u(y)
x − y dy.
For the proof see in [3, Lemma 7.16].
The following important inequality was derived in [4]:
Lemma 2.3. Let B = B(x, r) ⊂ Ω, u ∈ C ∞ (Ω), Γ ⊂ ∂Ω. Then
∫
∫
∇u(z)
inf
y − z dz ≤ ∇u(z)
x − z dz.
y∈Γ∩B
B
Lemma 2.4. If 0 < α < 2 and r > 0, then there exists a constant c 2 ,
c 2 ≤ ( 1 α−2
2)
, such that for each x ∈ R 2 ,
∫
u(y)
x − y dy ≤ c 2r α M(u)(x).
2−α
B(x,r)
For the proof we refer to [15, Lemma 2.8.3]. We will use this result for the
case α =1. Here, as usual, M(u) stands for the HardyLittlewood maximal
operator. Moreover, the following important theorem (the HardyLittlewood
theorem on Maximal Operator) will be used:
Theorem 2.1. If u ∈ L 2 (R 2 ), then there exist a constant c 3 > 0 such that
B
‖M(u)‖ 2 ≤ c 3 ‖u‖ 2 .
For the proof see e.g. in [15, Theorem 2.8.2].
6
3 The main results.
Consider the function
ρ ε (x) =
{
ρ(x), if x ∈ Π 1 ,
ρ(x)+(1− δ) ε, 2 if x ∈ Π 2.
(3.1)
Our first main result is the following pointwise inequality:
Theorem 3.1. Let u ε ∈ C ∞ (Ω, Γ ε ). Then there exist a constant C, C ≤ 4,
such that the pointwise inequality
u ε (x) ≤Cρ ε (x)M ρε(x)∇u ε χ B(x,ρε(x))(x) (3.2)
holds, where x ∈ Γ is satisfying that x − x = ρ(x).
Proof. Choose the point x ∈ Ω and denote by B := B(x, ρ ε (x)), where ρ ε (x)
is defined in (3.1).
Then B ∩ Γ ε ≠øforeachx ∈ Ω. Extend the function u ε in R 2 \ Ωby
reflecting it across the boundary. By applying Lemma 2.2 to the extended
u ε , we have for any y ∈ B ∩ Γ ε :
u ε (x) = u ε (x) − u ε (y) ≤u ε (x) − u εB  + u ε (y) − u εB ≤
⎛
⎞
∫
∫
≤ 2 ⎝
∇u ε (z) ∇u ε (z)
dz +
dz⎠ .
x − z y − z
B
B
(3.3)
Hence, by applying Lemma 2.3 to (3.3), we obtain that
∫
u ε (x) ≤2
B
∇u ε (z)
x − z
dz. (3.4)
Finally, taking into account Lemma 2.4 and (3.1), we have that
u ε (x) ≤2c 2 ρ ε (x) M ρε(x)(∇u ε )(x) =
{
4ρ(x)
( Mρ(x) (∇u
) ε ), x ∈ Π 1 ∩ B
(
(x, ρ(x)) ,
)
4 ρ(x)+ (1−δ) ε M (∇u (1−δ)
2 ρ(x)+ ε ε), x ∈ Π 2 ∩ B x, ρ(x)+ (1−δ) ε .
2
2
The proof is complete.
The next two theorems generalize some wellknown classical Hardytype
for p = 2 inequalities to a much more wide class of functions.
7
Theorem 3.2. Let ρ ε (x) be the function defined in (3.1) and 0 ≤ α0. Choose σ>0 and put v ε = u ε ρ σ ε . It is not difficult to derive
that
(
∇v ε  2 ∂uε
= ρ σ ε + σρ σ−1 ∂ρ ) 2 ( ) 2
ε
u ε + ρ 2σ ∂uε
ε ≤
∂x 1 ∂x 1 ∂x 2 (3.6)
≤ 2ρ 2σ
ε ∇u ε  2 +2σ 2 ρ 2σ−2
ε u 2 ε.
By now applying (3.5) with α =0tov ε , we obtain that
⎛
∫
∫
∫
ρ −2+2σ
ε u 2 ε dx ≤C ⎝
1 ρ 2σ
ε ∇u ε 2 dx + σ 2
Ω
Ω
8
Ω
ρ 2(σ−1)
ε
⎞
u ε  2 dx⎠ .
If 1 −C 1 σ 2 > 0, then we have that
∫
∫
ρ −2+2σ
ε u 2 2C 1
ε dx ≤
1 − 2C 1 σ 2
Ω
Ω
ρ 2σ
ε ∇u ε  2 dx.
Ω θ
Substituting σ by α 2
and denoting C 2C
1 by 1
1−2C 1
, we prove inequality (3.5).
σ 2
Finally, by approximating the functions from H 1 (Ω, Γ ε ) by smooth functions
belonging to C ∞ (Ω, Γ ε ), we can complete the proof.
Our final main result reads:
Theorem 3.3. Let ρ(x) =dist(x, Γ) and 0 ≤ α0 ( for all functions ) u ε ∈ H 1 (Ω, Γ ε ), where the constant
C(a, ε, δ, θ) =4+2 1−δ a 2
θ 2 δ ε2 (1 − δ) .
Proof. First we note that
∫ ( ) 2 ∫
∫
uε
dx ≤ 4 ∇u ε  2 dx ≤ 4
ρ
∇u ε  2 dx. (3.8)
Π 1
Π 1 Ω
Indeed, if u ε (0) = 0, then, according to the classical onedimensional inequality
(1.1), we have that
∫ a
u ′ ε dx 1 ≥ 1 ∫ a
( ) 2 uε
dx 1 .
0
4 0 x 1
Using this fact, we obtain that
∫
Π 1
∇u ε  2 dx ≥
∫Γ ε
∫ a
0
( ∂uε
∂x 1
) 2
dx ≥ 1 4
= 1 4
∫Γ ε
∫ a
∫
Π 1
0
(
uε
x 1
) 2
dx =
( ) 2 uε
dx.
ρ
In particular, the estimate (3.8) holds in Π θ 1 . The next step is to prove that
∫ ( ) 2 ∫
uε K(a, ε, δ)
dx ≤
ρ
θ 2 ∇u ε  2 dx. (3.9)
Ω
Π θ 2
9
Using the Friedrichs inequality (2.1) and taking into account the fact that
ρ(x) >θwhen x ∈ Π θ 2 , we obtain the following estimate:
∫ ( ) 2 uε
dx ≤ 1 ∫
∫
u 2 K(a, ε, δ)
ρ θ 2 ε dx ≤ ∇u
θ 2
ε  2 dx.
Π 2 Ω
Π θ 2
Now, summarizing the inequalities (3.8) and (3.9), we derive the desired
estimate: ∫ ( ) 2 ∫ ( ) 2 ∫ ( ) 2 uε
uε
uε
dx = dx + dx ≤
ρ
ρ
ρ
Ω θ
Π θ 1
Π θ 2
≤
(
4+
K(a, ε, δ)
θ 2
) ∫
Ω
∇u ε  2 dx.
We have derived the inequality (3.7) for the case α =0. The proof of (3.7)
for the case α>0 is identically to the proof of the second part of Theorem
3.2, so we omit the details.
4 Concluding remarks.
Remark 4.1. The condition δ =1in the definition (3.1) corresponds to
the case Γ ε =Γ. In this case the Hardytype inequality (3.5) becomes the
inequality (1.4) and the constant in (3.7) equals to the constant in the classical
Hardy inequality when the function is vanishing on the whole boundary.
Remark 4.2. If α =0, then (3.5) takes the form
∫ ( ) 2 ∫
uε (x)
dx ≤ C 1 ∇u ε (x) 2 dx, (4.1)
ρ ε (x)
Ω
Ω
while (3.7) becomes
∫ ( ) 2 ∫
uε (x)
dx ≤ C(a, ε, δ, θ)
ρ(x)
Ω
Ω θ
∇u ε (x) 2 dx.
We conjecture that these Hardytype inequalities holds also when p =2
is replaced by any p>1 but then another type of proof must be found.
Remark 4.3. In this paper we have succeeded to prove a weighted Hardytype
inequality in a fixed domain for functions vanishing on a part of the boundary.
10
We can see several open questions equipped with this result. For instance,
one interesting problem is to try to find a weighted Hardytype inequality for
perforated domains in the case when the size of perforation depends on the
small parameter.
Acknowledgments.
It is a pleasure for me to thank Professor LarsErik Persson and the referee
for various remarks and suggestions, which have improved the final version
of the paper. I also thank Professor Gregory A. Chechkin for interesting
discussions and help concerning the problem studied in this paper.
The work was partially supported by RFBR (project 090100353) and
by the grant of the President of Russian Federation for supporting “Leading
Scientific Schools” and by Luleå University of Technology (Sweden).
References
[1] G.A.Chechkin, Yu.O.Koroleva and L.E.Persson, On the precise
asymptotics of the constant in the Friedrich‘s inequality for functions,
vanishing on the part of the boundary with microinhomogeneous structure,
J. Inequal. Appl., 2007, Article ID 34138, 13 pages, 2007.
[2] G. A. Chechkin, Yu. O. Koroleva, A. Meidell and L. E. Persson, On the
Friedrichs inequality in a domain perforated nonperiodically along the
boundary. Homogenization procedure. Asymptotics in parabolic problems,
Russ. J. Math. Phys., 16 (2009), No.1, 1–16.
[3] D. Gilbarg and N. Trudinger, Elliptic Partial Differenrial Equations of
Second Order, SpringerVerlag, 1983.
[4] P. Hajlasz, Pointwise Hardy inequalities. Proc. Amer. Math. Soc. 127
(1999), 417–423.
[5] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge
Mathematical Library (Reprint of the 1952 edition ed.). Cambridge:
Cambridge University Press, 1988.
[6] I. S. Kac and M. G. Krein, Criteria for discreteness of the spectrum of a
singular string, Izv. Vyss. Uchebn. Zaved. Mat. (1958), No.2, 136–153.[in
Russian]
11
[7] J. Kinnunen and R. Korte, Characterizations for Hardy‘s Inequality.
Around the Research of Vladimir Maz’ya, International Mathematical
Series, (2010), No.11, 239–254.
[8] V. Kokilashvili, A. Meskhi and L.E. Persson, W eighted Norm Inequalities
For Integral Transforms With Product Kernals, Nova Science Publishers
Inc (United States), 2009.
[9] A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, 1985.
[10] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality.
About its History and some Related Results, Vydavetelsky Servis Publishing
House, Pilsen, 2007.
[11] A. Kufner and L.E. Persson, Weighted Inequalities of Hardy Type,World
Scientific, New JerseyLondonSingaporeHong Kong, 2003.
[12] V. G. Maz’ja, Sobolev spaces. Translated from the Russian by T. O.
Shaposhnikova. Springer Series in Soviet Mathematics. SpringerVerlag,
Berlin, 1985.
[13] J. Nečas, Sur une méthode pour résourde les équations aux dérivées partielle
du type elliptique, voisine de la variationelle, Ann. Scuola Norm.
Sup. Pisa, 16 (1962), 305–362.
[14] B. Opic and A. Kufner, HardyType Inequalities, Longman, Harlow,
1990.
[15] W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics
120, SpringerVerlag, 1989.
[16] A. Wannebo, Hardy Inequalities and Imbedding in Domains Generalizing
C 0,λ Domains, Proc.Amer.Math.Soc.,122 (1994), 11811190.
Yu.O.Koroleva,
Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
&
Department of Mathematics
Luleå University of Technology
SE971 87 Luleå, Sweden
korolevajula@mail.ru
Paper F
On the weighted Hardytype inequality in a
domain with perforation along the boundary.
Yu.O. Koroleva
Key words and phrases. Partial Differential Equations, Functional Analysis,
Spectral Theory, Homogenization Theory, Hardytype Inequalities
AMS 2010 Subject Classification. 39A10, 39A11, 39A70,
39B62, 41A44, 45A05
Abstract
In present paper we derive threedimensional weighted Hardytype
inequality in a cube for the class of functions from Sobolev space H 1
having zero trace on small holes distributed periodically along the
boundary .
1 Introduction.
Integral inequalities are very important for different applications. Fore example,
they are often used for deriving some estimates for operator norms,
for proving some embedding theorems, etc.
Let Ω ⊆ R n . Weighted Hardytype inequality is the integral inequality as
follows:
⎛
⎞ 1 ⎛
⎞ 1
∫
q ∫
p
⎝ U(x) q V (x) dx⎠
≤C⎝
∇U(x) p W (x) dx⎠
, (1.1)
Ω
Ω
where U(x) ∈ C0 ∞ (Ω), V(x) ≥ 0,W(x) ≥ 0, 1 ≤ p, q < ∞, and the constant
C does not depend on the function U. There are several results concerning
weighted Hardytype inequalities (see e.g. the books [15], [16] and [20] and
the references given there).
The aim of this paper is to derive the Hardytype inequality
∫
∫
U(x) 2 ρ α−2 (x) dx ≤ C ∇U(x) 2 ρ α (x) dx (1.2)
Ω
Ω
where Ω is bounded and has nontrivial microstructure. More precisely, we
assume that Ω is a cube with perforation along a part of the boundary and
that the weight function ρ decreases to zero as x approaches the part of the
1
oundary which is associated with the perforation. It should be mentioned
that results in this direction are completely new in the theory of Hardytype
inequalities. In particular, it gives us possibility to use ideas developed
within the homogenization theory to obtain estimates for the best constant in
different Hardytype inequalities. The first step in this direction was recently
taken in [12], where the inequality (1.2) was proved under the assumption
that the function U vanishes on small alternating pieces of a part of the
boundary.
In present paper we derive the inequality (1.2) under the assumption that
Ω is a cube with perforation along a part of the boundary.
Such a result is completely new in the theory of Hardytype inequalities
and it gives us possibility to apply the tools of homogenization theory to
obtain the estimates for the best constant in the Hardytype inequality in a
perforated domain.
The paper is organized as follows: in Section 2 we give some necessary
definitions and formulate the main results which are proved in Section 3
and Section 5. The proof of main theorem from Section 3 is based on some
auxiliary lemmas which are proved in Section 4.
2 Statement of the problem and
the main result.
Let Ω ⊂ R 3 be the cube
{0
Figure 1: The domain Ω ε .
Remark 2.1. Without loss of generality we assume that U ε is extended to be
U ε ≡ 0inB ε .
Analogously,
H 1 (Ω, Γ) = {U ∈ H 1 (Ω) : U Γ =0},
where U Γ is the trace of the function U. Let x ∈ Ω ε . Define the function
ρ(x) :=dist(x, Γ). Ourfirstmainresultreadsasfollows.
Theorem 2.1. Suppose that U ε ∈ H 1 (Ω, Γ ε ). Then the Friedrichs inequality
∫
∫
Uε 2 dx ≤ K ε ∇U ε  2 dx (2.1)
Ω
Ω
holds, where
K ε = K 0 + ε λ 1
K 2 0
)
+ o
(ε 3 2 −μ , 0
Theorem 2.2. Let ρ(x) =dist(x, Γ) and 0 ≤ α 0 is a constant in a Friedrichs inequality for functions from H 1 (Ω, Γ).
The third important result obtained in this paper is the following Theorem.
Theorem 2.3. Assume that ρ(x) = dist(x, Γ), 0 ≤ α ≠ 1,
U ε ∈ H 1 (Ω, Γ ε ). Then there is a function M ε (x 2 ,x 3 ),
‖M ε ‖ L2 (Γ) ≤C √ ε, (2.4)
such that a Hardytype inequality
∫
U ε − M ε  2 ρ α−2 dx ≤
holds true.
Ω
∫
4
(α − 1) 2
Ω
∇U ε  2 ρ α dx (2.5)
Remark 2.2. All results can be extended to other domains with sufficiently
smooth boundary.
3 Proof of Theorem 2.1.
The proof is built up of a sequence of results and facts, which are of independent
interest and we therefore write them as Lemmas or Remarks. Except
for Lemma 3.2, the lemmas are proved in the next section. Note that the
validity of (2.1) and convergence K ε to K 0 as ε → 0 are proved in paper [11]
(see also [13] for an aperiodical case). In this section we derive asymptotics
(2.2) for the constant K ε in Friedrichs inequality (2.1) via the method of
matching of asymptotic expansions.
Consider the following spectral problem
⎧
⎪⎨ −Δu ε = λ ε u ε in Ω ε ,
u ε =0 onΓ ε ,
(3.1)
⎪⎩ ∂u ε
=0 on∂Ω. ∂ν
4
Here and further we denote by ν the unit outward vector.
The problem ⎧
⎪⎨ −Δu 0 = λ 0 u 0 in Ω,
u
⎪ 0 =0 onΓ,
(3.2)
⎩ ∂u 0
=0 on∂Ω \ Γ ∂ν
is limit problem for (3.1). This fact could be established analogously like it
was done in [5] for the twodimensional case. In particular, the convergence
of any eigenvalue λ ε of the problem (3.1) to the corresponding eigenvalue λ 0
of the problem (3.2) as ε → 0 was established. Moreover, the convergence of
the corresponding eigenfunctions in norms of Sobolev space H 1 was derived.
Our aim is to construct the first two terms of asymptotic expansion for
simple eigenvalues for the spectral problem (3.1).
Define the sets
Π={ξ 1 > 0, − 1 2
Figure 2: Cell of periodicity.
where 0
while we use the formula
u ε (x) ≈ û ε = u 0 (x)+εu 1 (x)+ε 2 u 2 (x) (3.8)
for asymptotics of u ε .
We have that u 0 ∈ C ∞ (Ω), see [5]. If we substitute the expansions (3.7)
and (3.8) in the spectral problems (3.1) and collect term of the same order
of ε, then by taking into account (3.2), we obtain the expansion
u 0 (x) =α 1 0 (x 2,x 3 )x 1 + O(x 3 1 ) (3.9)
as x 1 → 0, where
∣
α0 1 = ∂u ∣∣∣∣x1 [
0
∈ C ∞ − 1 ∂x 1 2 , 1 ] [
× − 1 2 2 , 1 ]
2
=0
(3.10)
and
∂ 2j+1 α 1 0
∂x 2j+1
2
(
± 1 )
2 ,x 3 =0,
∂ 2j+1 α 1 0
∂x 2j+1
3
(
x 2 , ± 1 )
=0, (3.11)
2
where j =0, 1, 2,....
We choose the functions u 1 and u 2 satisfying the boundaryvalue problems
⎧
⎪⎨
−Δu 1 = λ 0 u 1 + λ 1 u 0 in Ω,
∂u 1
=0 on∂Ω \ Γ,
(3.12)
⎪⎩ ∂ν
u 1 = α1 0 on Γ,
⎧
⎪⎨
⎪⎩
−Δu 2 = λ 0 u 2 + λ 1 u 1 + λ 2 u 0 in Ω,
∂u 2
=0 on∂Ω \ Γ,
∂ν
u 2 = α2 0 on Γ,
(3.13)
where α1(x 0 2 ,x 3 ),α2(x 0 2 ,x 3 ) are unknown functions which will be defined further.
Remark 3.1. Equations of boundaryvalue problems (3.12), (3.13) and boundary
conditions (except condition on Γ) are the result of substituting the expansions
(3.7) and (3.8) in (3.1) and collecting terms of the same order of
ε.
The validity of the following Lemma can be established by using the same
technique as in the proof of the analogous result in [1]. We omit the details.
7
Lemma 3.3. Assume that α 0 1,α 0 2 ∈ C ∞ [ − 1 2 , 1 2]
×
[
−
1
2 , 1 2]
and that
odd derivatives of functions α 0 1 and α0 2 with respect to x 2,x 3 vanish as
x 2 = ± 1 2 , x 3 = ± 1 2 . Then there exist constants λ 1, λ 2 and the functions
u 1 (x),u 2 (x) ∈ C ∞ (Ω) which are the solutions of problems (3.12) and
(3.13) respectively. Moreover, λ 1 satisfies
λ 1 = −
1 1
∫2 ∫2
α 0 1 (x 2,x 3 )α 1 0 (x 2,x 3 ) dx 2 dx 3 . (3.14)
− 1 − 1 2 2
By using the Teylor expansion, we obtain that
u 1 (x) =α1(x 0 2 ,x 3 )+α1(x 1 2 ,x 3 )x 1 + O(x 2 1),
u 2 (x) =α2 0 (x 2,x 3 )+O(x 1 )
(3.15)
as x 1 → 0whereα1 1 ∈ C [ ∞ − 1, [ 1
2 2]
× −
1
, 1 2 2]
and
(
± 1 )
2 ,x 3 =0,
∂ 2j+1 α 1 1
∂x 2j+1
2
∂ 2j+1 α 1 1
∂x 2j+1
2
(
x 2 , ± 1 )
=0, (3.16)
2
where j =0, 1, 2,... due to (3.12).
Taking into account Remark 3.1 and Lemma 3.3 we conclude that the
following Lemma holds.
Lemma 3.4. Assume that α1 0,α0 2 ∈ [ C∞ − 1, [ 1
2 2]
× −
1
, 1 2 2]
and odd derivatives
of functions α1 0 and α2 0 with respect to x 2 and x 3 vanish at points ( ± 1,x )
2 3
and ( x 2 , ± 2) 1 . Then ûε ∈ C ∞ (Ω) and the formulas
are valid.
⎧
⎨
⎩
−Δû ε = ̂λ ε û ε + O(ε 3 ) in Ω,
∂û ε
=0 on ∂Ω \ Γ
∂ν
We construct another interpolation for function u ε in a small neighborhood
of Γ (inner expansion) since the function û ε (x) does not satisfies to the
boundary conditions of the problem (3.1) on Γ and on Γ ε .
Formulas (3.8), (3.9) and (3.15) lead to the following:
û ε (x) =α0 1 (x 2,x 3 )x 1 + ε(α1 0 (x 2,x 3 )+α1 1 (x 2,x 3 )x 1 )+ε 2 α2 0 (x 2,x 3 )
+ O(x 3 1 + εx 2 1 + ε 2 x 1 ) as x 1 → 0.
8
Put ξ 1 = x 1
ε
, then we conclude that
û ε (x) =εV 1 (ξ 1 ; x 2 ,x 3 )+ε 2 V 2 (ξ 1 ; x 2 ,x 3 )
+ O(x 3 1 + εx 2 1 + ε 2 x 1 ) as εξ 1 → 0,
(3.17)
where
V 1 (ξ 1 ; x 2 ,x 3 )=α0 1 (x 2,x 3 )ξ 1 + α1 0 (x 2,x 3 ),
V 2 (ξ 1 ; x 2 ,x 3 )=α1(x 1 2 ,x 3 )ξ 1 + α2(x 0 (3.18)
2 ,x 3 ).
According to the method of matching of asymptotic expansions we conclude
that the internal expansion have to be of the following structure in a
neighborhood of Γ:
u ε (x) ≈ ̂v ε (x) =εv 1 (ξ; x 2 ,x 3 )+ε 2 v 2 (ξ; x 2 ,x 3 ) , (3.19)
where ξ = x ε , v q (ξ; x 2 ,x 3 ) ∼ V q (ξ 1 ; x 2 ,x 3 ) as ξ 1 → +∞. (3.20)
Here x 2 ,x 3 are so called “slow” variables while ξ is “fast” variable.
The equation of the problem (3.1) with respect to variables (ξ; x 2 ,x 3 )has
the following form:
∂2 u ε
∂2 u ε
−ε −2 Δ ξ u ε − 2ε −1 − 2ε −1 − ∂2 u ε
∂x 2 ∂ξ 2 ∂x 3 ∂ξ 3 ∂x 2 2
− ∂2 u ε
∂x 2 3
= λ ε u ε . (3.21)
Boundary conditions on the lateral surface of the cell of periodicity Π except
γ are
∂u ε
∂ν = ∂u ±ε−1 ε
± ∂u ε
=0, (3.22)
∂ξ 2 ∂x 2
∂u ε
∂ν = ±ε−1 ∂u ε
∂ξ 3
± ∂u ε
∂x 3
=0, (3.23)
and on γ is
∂u ε
∂ν = ∂u −ε−1 ε
− ∂u ε
=0. (3.24)
∂ξ 1 ∂x 1
Remark 3.2. We construct further the internal expansion for (3.19) as 1
periodic function with respect to ξ 2 and ξ 3 .
Rewriting the equation and boundary conditions in ξ variables (see (3.21)
– (3.24)), substituting (3.19) and (3.7) in (3.1), finally, we equate terms at ε q
9
corresponding to the same q. Taking into account (3.20), (3.18) and Remark
3.2, we get the following boundaryvalue problem for v 1 :
⎧
Δ ξ v 1 =0 inΠ\B,
⎪⎨
⎪⎩
v 1 =0 on∂B,
∂v 1
∂ξ 1
=0 onγ,
∂v 1
∂ξ 2
(ξ; ± 1 2 ,x 3)=0 asξ 2 = ± 1 2 ,
∂v 1
∂ξ 3
(ξ; x 2 , ± 1 2 )=0 asξ 3 = ± 1 2 ,
v 1 ∼ V 1 as ξ 1 → +∞
(3.25)
and for v 2 :
⎧
−Δ ξ v 2 =2 ∂2 v 1
∂x 2 ∂ξ 2
+2 ∂2 v 1
∂x 3 ∂ξ 3
in Π\B,
v 2 =0 on∂B,
⎪⎨ ∂v 2
∂ξ 1
=0 onγ,
∂v 2
∂ξ 2
(ξ; ± 1 ,x 2 3)=− ∂v 1
∂x 2
(ξ; ± 1 ,x 2 3) as ξ 2 = ± 1 , 2
∂v 2
∂ξ ⎪⎩ 3
(ξ; x 2 , ± 1)=− ∂v 1
2 ∂x 3
(ξ; x 2 , ± 1) as ξ 2 3 = ± 1,
2
v 2 ∼ V 2 as ξ 1 → +∞.
(3.26)
Due to the boundaryvalue problems (3.25) and (3.26) we conclude that
the following Lemma is valid.
Lemma 3.5. Assume that the solutions to problems (3.25) and (3.26) exist
and are 1periodic functions with respect to ξ 2 and ξ 3 . Then the functions ̂v ε ,
̂λ ε , which are given by (3.19) and (3.7) respectively, satisfy to the following
formulas for each sufficiently small h>0 :
−Δ̂v ε = ̂λ ε̂v ε + ̂F v ε
in Ω ε ∩{x 1
Now we study the solvability of the boundaryvalue problem (3.25) and
determine the formula for function α1 0(x 2,x 3 ).
It should be noted that the function X, defined in (3.3), can be extended 1
periodically with respect to ξ 2 and ξ 3 . We save the same notation for extended
function. Put
v 1 (ξ; x 2 ,x 3 )=α0(x 1 2 ,x 3 )X(ξ). (3.29)
Due to (3.4) this function has the following asymptotics
v 1 (ξ; x 2 ,x 3 )=α 1 0(x 2 ,x 3 )ξ 1 + α 1 0(x 2 ,x 3 )C(B)+O(e −2πξ 1
) (3.30)
as ξ 1 → +∞. Consequently, by using Lemma 3.1 and assuming that
α 0 1 (x 2,x 3 )=α 1 0 (x 2,x 3 )C(B), (3.31)
we conclude that the function v 1 is a solution of (3.25). Moreover,
v 1 (ξ; x 2 ,x 3 )=V 1 (ξ 1 ; x 2 ,x 3 )+O(e −2πξ 1
) as ξ 2 → +∞, (3.32)
α1 0 ∈ C [ ∞ − 1 , ] [ 1
2 2 × −
1
2 2]
and due to (3.11):
(
∂ 2j+1 α1
0
∂x 2j+1 ± 1 )
(
2 ,x ∂ 2j+1 α1
0
3 =0,
2
∂x 2j+1 x 2 , ± 1 )
=0,
2
3
j =0, 1, 2,...
Summarizing all results, we deduce that the condition of solvability for
(3.25) let us to determine the function α1 0(x 2,x 3 ) in boundaryvalue problem
(3.12) which satisfies the conditions of Lemma 3.3. On the other hand, the
formula (3.6) follows directly from (3.14) and (3.31).
Note that due to (3.11) and (3.29) the formula
∂v 1
∂x 2
(ξ; x 2 ,x 3 )=0,
∂v 1
∂x 3
(ξ; x 2 ,x 3 )=0, (3.33)
holds as x 2 = ± 1 ,x 2 3 = ± 1 , respectively.
2
Now we begin to study the problem (3.26).
Lemmas are proved in Section 4.
The following important
Lemma 3.6. The boundaryvalue problem
⎧
Δ ˜X 2 = ∂X
∂ξ 2
in Π\B,
⎪⎨
˜X 2 =0 on ∂B,
∂ ˜X 2
∂ξ 1
=0 on γ,
˜X 2 =0 as ξ 2 = ±
⎪⎩
1 2
˜X 2 =0 as ξ 3 = ± 1 2
(3.34)
11
has a solution which is odd with respect to ξ 2 and even with respect to ξ 3 and
satisfies the following asymptotics
˜X 2 (ξ) =O(e −ξ 1
) as ξ 1 →∞. (3.35)
Lemma 3.7. The boundaryvalue problem
⎧
Δ ξ ˜X3 = ∂X
∂ξ 3
in Π\B,
⎪⎨
˜X 3 =0 on ∂B,
∂ ˜X 3
∂ξ 1
=0 on γ,
˜X 3 =0 as ξ 2 = ±
⎪⎩
1 2
˜X 3 =0 as ξ 3 = ± 1 2
(3.36)
has a solution which is even with respect to ξ 2 , odd with respect to ξ 3 and has
the following asymptotics
˜X 3 (ξ) =O(e −ξ 1
) as ξ 1 →∞. (3.37)
Due to 1periodicity of the righthand side of the equation in problem
(3.34) and the boundary conditions we can extend ˜X 2 and ˜X 3 1periodically
with respect to ξ 2 and ξ 3 . We will use the same notation for the extended
functions.
Put
v 2 (ξ; x 2 ,x 3 )=α1 1 (x 2,x 3 )X(ξ) − 2 ∂α1 0
(x 2 ,x 3 )
∂x ˜X 2 (ξ) − 2 ∂α1 0
(x 2 ,x 3 )
2 ∂x ˜X 3 (ξ).
3
This function is 1periodic with respect to ξ 2 and ξ 3 and in view of (3.4),
(3.35) and (3.37) it has asymptotics
v 2 (ξ; x 2 ,x 3 )=α 1 1 (x 2,x 3 )ξ 1 + C(B)α 1 1 (x 2,x 3 )+O(ξ 1 e −ξ 1
) (3.38)
as ξ 1 →∞. Hence, taking into account Lemmas 3.1, 3.6, 3.7 and formulas
(3.30), (3.11) and (3.33), we deduce that v 2 is a solution of (3.26) if
moreover,
α 0 2 (x 2,x 3 )=α 1 1 (x 2,x 3 )C(B),
v 2 (ξ; x 2 ,x 3 )=V 2 (ξ 1 ; x 2 ,x 3 )+O(ξ 1 e −ξ 1
)asξ 1 →∞, (3.39)
α2 0 ∈ C [ ∞ − 1, 1 2 2]
and due to (3.16):
(
∂ 2j+1 α2
0
∂x 2j+1 ± 1 )
2 ,x ∂ 2j+1 α2
0
3 =0,
2
∂x 2j+1
3
12
(
x 2 , ± 1 )
=0, j =0, 1, 2,...
2
Hence, the solvability conditions for boundaryvalue problem (3.26) determine
the function α2 0(x 2,x 3 ) which satisfies the conditions of Lemma 3.3.
Note that due to (3.16) and the boundary conditions ˜X 2 = ˜X 3 =0on
∂Π
(
∂v 2
ξ 1 , ± 1 )
∂x 2 2 ,ξ 3; x 2 ,x 3 =0asx 2 = ± 1 2 ,
(
∂v 2
ξ 1 ,ξ 2 , ± 1 )
∂x 3 2 ; x 2,x 3 =0asx 3 = ± 1 (3.40)
2 ,
consequently, taking into account (3.40) and (3.27) we obtain that
∂̂v
(
ε x
)
∂ν ε ; x 2,x 3 =0onΓ∪ ((∂Ω \ Γ) ∩{x 1
where
moreover,
The estimate
‖ ̂f ε ‖ L2 (Ω ε) = O(ε 3 2 β ), (3.42)
lim ‖Ûε‖ 0 ≥ 1. (3.43)
ε→0
‖Ûε‖ L2 (Ω ε) ≤ C ‖ ̂f‖ L2 (Ω ε)
∣
∣λ ε − ̂λ
∣ ∣∣
ε
can be proved exactly in the same way like it was done in paper [4] for the
twodimensional case. Here Ûε is a solution of the boundaryvalue problem
(3.41) and the constant C does not depend on ε. This fact together with
(3.42) and (3.43) give us the following formula:
λ ε − ̂λ ε  = O(ε 3 2 β ). (3.44)
The formula (3.5) holds due to (3.7) and (3.44) since β is an arbitrary number
in the interval (0, 1). The proof of Lemma 3.2 is complete.
Taking into account the variational definition of K ε and the asymptotics
(3.5) we get the formula (2.2). The proof of Theorem 2.1 is complete.
14
4 Proofs of the Lemmas
Proof of Lemma 3.8. Taking into account (3.30) and (3.38), we have
∫ [ (
‖ ̂F ∂
ε v 2 ‖2 L 2 (Ω ε∩{x 1
Proof of Lemma 3.10. The validity of (3.43) is obvious. The function Ûε
satisfies to the boundary conditions of problem (3.41) due to Lemmas 3.4,
3.5 and formula (3.40). By applying the operator −(△ + ̂λ ε )toÛε we get
where
̂f ε = I 1 + I 2 + I 3 ,
I 1 = − χ β (△û ε + ̂λ ε û ε ),
I 2 = − (1 − χ β )(△̂v ε + ̂λ ε̂v ε )=−(1 − χ β ) ̂F ε v ,
I 3 =(̂v ε − û ε )△χ β +2∇χ β ∇ x (̂v ε − û ε )
(
=ε −2β χ ′′ x1
)
(
(̂v
ε β ε − û ε )+2ε −β χ ′ x1
) ∂
(̂v
ε β ε − û ε ) .
∂x 1
Using Lemma 3.4, we obtain that
⎛
∫
‖I 1 ‖ L2 (Ω ε) = ⎝
⎞
I1(x) 2 dx⎠
1
2
= ( O(ε 6 ) ) 1 2
= O(ε 3 ). (4.1)
Ω ε
Lemma 3.8 together with the definition of function χ β give the following
asymptotics:
⎛
∫
‖I 2 ‖ L2 (Ω ε) = ⎝
⎞
I2 2 (x) dx⎠
1
2
=
⎛
⎜
⎝
∫
⎞
I2(x) 2 ⎟
dx⎠
1
2
=
Ω ε
= ( O(ε 3β ) ) 1 2
= O
(ε 3 2 β )
.
Ω ε∩{x 1
Lemma 4.1. Assume that e δ 0ξ 1
F ∈ L 2 (Π \ B) e δ 0ξ 1
H ∈ L 2 (∂Π),
K ∈ H 1 2 (∂B) and δ 0 > 0. Then there exist a weak solution of the following
boundaryvalue problem:
⎧
−ΔZ = F in Π \ B
⎪⎨
Z = K on ∂B,
⎪⎩ ∂Z
∂ν = H
This solution could be given by the formula
on ∂Π.
Z(ξ) =C + Z ′ (ξ),
where C is a constant, e δξ 1
Z ′ ∈ H 1 (Π\B) and δ is an arbitrary number
satisfying to the conditions δ ≤ δ 0 and δR}, γ R = {ξ ∈ Π,ξ 1 = R}, y R = Y ∣ γR
. Obviously,
the function Y is also a unique classical bounded solution of the following
boundaryvalue problem
⎧
ΔY =0 inΠ R ,
⎪⎨ ∂Y
∂ξ 1
= y R on γ R ,
∂Y
∂ξ 2
=0 asξ 2 = ± 1 2
⎪⎩
,
∂Y
∂ξ 3
=0 asξ 3 = ± 1 2
when R is sufficiently large number. Hence, taking into account that Y is an
even function with respect to ξ 2 and ξ 3 we deduce
Y (ξ) =C(B)+O(e −2πξ 1
) as ξ 1 → +∞,
17
consequently, (3.4) holds.
Let us prove the formula
C(B) =
∫
Π\B
∇Y  2 dξ + B > 0. (4.4)
Denote by Π R =Π∩{ξ 1
we get
∫
∂B
ξ 1
∂ξ 1
∂ν B
ds ξ = B, (4.9)
where ν B is an outward normal vector to B. The formula (4.4) follows from
(4.8) and (4.9).
ProofofLemma3.6. The proof of this Lemma is based on the following
Lemma from [18].
Lemma 4.2. Assume that e δ 0ξ 1
F ∈ L 2 (Π \ B) and δ 0 > 0. Then there exist
the unique solution of the following boundaryvalue problem:
⎧
−ΔZ = F in Π \ B
⎪⎨
Z =0 on ∂B ∪ ∂Π\γ,
⎪⎩ ∂Z
=0 on γ.
∂ν
The following embedding e δξ 1
Z ∈ H 1 (Π\B) holds for Z, moreover, δ is an
arbitrary number satisfying the conditions δ ≤ δ 0 and δ0. Choose σ>0 and put V ε = U ε ρ σ . It is
not difficult to derive
( ) 2
∇V ε  2 ∂Uε
= ρ σ σ−1
∂ρ
+ σρ U ε + ρ 2σ ∇ x2 x
∂x 1 ∂x
3
U ε  2 ≤
1 (5.2)
≤ 2ρ 2σ ∇U ε  2 +2σ 2 ρ 2σ−2 Uε 2 .
19
By applying (5.1) to V ε , we obtain that
⎛
∫
ρ −2+2σ
ε Uε 2 dx ≤ 2K ∫
∫
0
⎝ ρ 2σ ∇U
θ 2
ε  2 dx + σ 2
Ω θ
Ω
Ω
⎞
ρ 2(σ−1) Uε 2 dx ⎠ .
If 1 − 2K 0
σ 2 > 0, then
θ 2
∫
∫
ρ −2+2σ Uε 2 dx ≤ 2K 0
θ 2 − 2K 0 σ 2
Ω θ Ω
ρ 2σ ∇U ε  2 dx.
Finally, denoting by α the constant σ , we obtain (2.3) with
2
C(θ, α) =
4K 0
2θ 2 − K 0 α 2 ,
α < √
2θ
√
K0
.
Proof of Theorem 2.3. We assume at first that U ε ∈ C ∞ (Ω, Γ ε ). Fix the
variables x 2 ,x 3 and use at first the following onedimensional Hardytype
inequality:
∫ 1
v 2 ∫1
(x 1 ) 4
x 2−α dx 1 ≤
x α
1
(α − 1) 2 1 (v′ ) 2 dx 1 ,
0
where v ∈ AC[0, 1] and v(0) = 0. By applying this inequality to the function
v(x 1 )=U ε (x 1 , ·, ·) − U ε (0, ·, ·), we obtain that
∫ 1
∫1
(U ε (x 1 ,x 2 ,x 3 ) − U ε (0,x 2 ,x 3 )) 2 ρ α−2 4
dx 1 ≤
ρ α ∇U
(α − 1) 2 ε  2 dx 1 .
0
0
(5.3)
Denote by M ε (x 2 ,x 3 ):=U ε (0,x 2 ,x 3 ). By integrating the inequality (5.3)
with respect to x 2 and x 3 , we deduce that
∫
∫
(U ε − M ε ) 2 ρ α−2 4
dx ≤
ρ α ∇U
(α − 1) 2 ε  2 dx. (5.4)
Ω
Finally, we approximate the functions U ε ∈ H 1 (Ω, Γ ε ) by smooth functions
from C ∞ (Ω, Γ ε ) and conclude that (5.4) is valid also for U ε ∈ H 1 (Ω, Γ ε ). The
next step is to derive the estimate (2.4).
0
Ω
20
There exists a sequence of functions Uε k ∈ C ∞ (Ω, Γ ε ) such that Uε k converges
to U ε in H 1 as k →∞. Denote by Mε k := U ε k(0,x 2,x 3 ). Consequently,
Mε
k converges to M ε as k →∞in H 1 2 . Choose a number K such that
‖M ε − Mε k ‖ L 2 (Γ) ≤ C √ ε for any k>K. (5.5)
Let us prove that there is a constant C such that
‖M k ε ‖ L 2 (Γ) ≤ C √ ε. (5.6)
Suppose that there exists a subsequence ε n ,n→∞, such that
Hence, max
Γ
‖M k ε n
‖ 2 L 2 (Γ) >n2 ε n .
M k ε n
 >n √ ε n . Consequently,
∫
L 2 (Ω∩{x 1 ≤ε n})
⎛
∣
∣ ∇U
k 2
εn
dx > εn ⎝ max M ⎞
ε k Γ n

⎠
ε n
2
>n 2 ,
that is U k ε does not belong to H 1 . This contradiction proves (5.6).
taking into account (5.6) and (5.5) we deduce that
‖M ε ‖ L2 (Γ) ≤‖M ε − M k ε ‖ L 2 (Γ) + ‖M k ε ‖ L 2 (Γ) ≤ 2C √ ε.
The last estimate with 2C = C proves (2.4).
Acknowledgments.
It is a pleasure for me to thank Professors Gregory A. Chechkin, LarsErik
Persson and Professor Peter Wall for various remarks and suggestions, which
have improved the final version of the paper.
The work was partially supported by RFBR (project 090100353) and
by the program “Leading Scientific Schools” (HIII7429.2010.1) and by Luleå
University of Technology (Sweden).
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Department of Differential Equations,
Faculty of Mechanics and Mathematics,
Moscow Lomonosov State University,
Moscow 119991, Russia
&
Department of Mathematics
Luleå University of Technology
SE971 87 Luleå, Sweden
korolevajula@mail.ru