Polyharmonic boundary value problems

2.2 Higher order Sobolev spaces 29
(u,v)H m 0 :=
⎧
⎪⎨ ∆
Ω
⎪⎩
k u ∆ k vdx if m = 2k,
∇(∆
Ω
k u) · ∇(∆ k v)dx if m = 2k + 1,
(2.10)
and the corresponding norm
uH m 0 :=
⎧
⎨ ∆ k uL2 if m = 2k,
⎩
∇(∆ k u)L2 if m = 2k + 1.
(2.11)
For general p ∈ (1,∞), one has the choice of taking the Lp-version of (2.11) or the
equivalent norm
:= D m uLp. u W m,p
0
Thanks to these norms, one may define the above spaces in a different way.
Theorem 2.1. If Ω ⊂ R n is a bounded domain, then
W m,p
0 (Ω) = the closure of C∞ c (Ω) with respect to the norm .W m,p
= the closure of C∞ c (Ω) with respect to the norm . m,p
W .
0
Theorem 2.1 follows by combining interpolation inequalities (see [1, Theorem
1,p
4.14]) with the classical Poincaré inequality ∇uLp ≥ cuLp for all u ∈ W0 (Ω).
If Ω is unbounded, including the case where Ω = Rn , we define
u D m,p (Ω) := D m u L p (Ω),
D m,p (Ω) := the closure of C ∞ c (Ω) with respect to the norm .D m,p,
and, again, let W m,p
0 (Ω) denote the closure of C∞ c (Ω) with respect to the norm
.W m,p. In this unbounded case, a similar result as in Theorem 2.1 is no longer true
since although W m,p
0 (Ω) ⊂ D m,p (Ω), the converse inclusion fails. For instance, if
Ω = Rn , then W m,p
0 (Rn ) = W m,p (Rn ), whereas the function u(x) = (1 + |x| 2 ) (1−n)/4
belongs to D 1,2 (Rn ) but not to H1 0 (Rn ) = H1 (Rn ).
Theorem 2.1 states that, when Ω is bounded, the space Hm 0 (Ω) is a Hilbert space
when endowed with the scalar product (2.10). The striking fact is that not only
all lower order derivatives (including the derivative of order 0!) are neglected but
also that some of the highest order derivatives are dropped. This fact has a simple
explanation since
(u,v)H m 0 =
D
Ω
m u · D m vdx for all u,v ∈ H m 0 (Ω). (2.12)
One can verify (2.12) by using a density argument, namely for all u,v ∈ C ∞ c (Ω).
And with this restriction, one can integrate by parts several times in order to obtain
(2.12). The bilinear form (2.10) also defines a scalar product on the space D m,2 (Ω)
whenever Ω is an unbounded domain. We summarise all these facts in