# A nonlinear elliptic equation with rapidly oscillating ... - IOS Press A nonlinear elliptic equation with rapidly oscillating ... - IOS Press

Asymptotic Analysis 28 (2001) 279–307 279IOS PressA nonlinear elliptic equation with rapidlyoscillating boundary conditionsJuan DávilaDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08903, USAE-mail: davila@math.rutgers.edu1. Introduction1.1. MotivationLet Ω ⊂ R n , n 2, be a bounded, smooth domain, and consider a partition {Γ 1 , Γ 2 } of the boundary∂Ω,thatisΓ 1 ∪ Γ 2 = ∂Ω and Γ 1 ∩ Γ 2 = ∅, with Γ 1 ≠ ∅.Consider the problem⎧−∆u = λf(u) inΩ,⎪⎨u = 0 onΓ 1 ,⎪⎩ ∂u∂ν = 0 onΓ 2,where ν is the unit outward normal vector to ∂Ω, λ is a positive parameter, and f :[0,∞) → [0, ∞)isaC 1 nondecreasing, strictly convex function, with f(0) > 0and∫ ∞0ds< ∞. (2)f(s)Typical examples are f(u) = e u and f(u) = (1 + u) p where p>1. This type of nonlinear problemsarises, for example, from a model of exothermic reaction, and was originally formulated on a disk in R 2with zero boundary condition. Barenblatt et al.  introduced a modification of the original model byconsidering a mixed boundary condition as in (1).The case of a zero Dirichlet condition has been well studied, see, for example, Fujita , Gelfand, Brezis et al. , Brezis , Martel , Brezis and Vázquez . Some of the basic propertiesdescribed in these works still hold for (1): there is a value λ ∗ ∈ (0, ∞) such that for λλ ∗ (1) has no solution. For λ = λ ∗ there is a unique solution u ∗ (see Section3.3 and also Proposition 1.5 below). We call λ ∗ the extremal parameter associated to Γ 1 , Γ 2 ,andu ∗the extremal solution. In the original model, λ is a constant depending on physical parameters, and therelevance of λ ∗ is that a nonexplosive reaction is possible only if λ λ ∗ .We consider now a family {Γ1 ε, Γ 2 ε} ε>0 of partitions of the boundary, that is, Γ1 ε, Γ 2 ε ⊂ ∂Ω, Γ 1 ε∪Γ 2 ε =∂Ω, Γ1 ε ∩ Γ 2 ε = ∅, and we assume |Γ 1 ε | > 0forallε. Hereε is a positive index approaching zero, andwe denote by λ ∗ ε the corresponding extremal parameter. There are several ways in which we want this(1)0921-7134/01/\$8.00 © 2001 – IOS Press. All rights reserved

280 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsfamily to behave as ε → 0, but the general idea is that the partition Γ1 ε, Γ 2 ε “becomes finer” as ε → 0.For example we can consider the case in which Ω is the unit disk in R 2 , ∂Ω is subdivided in segmentsof length ε, and we impose homogeneous Dirichlet and Neumann conditions on alternate segments. Inthis particular case, Barenblatt suggested to study the asymptotic behavior of the extremal parameters λ ∗ εas ε → 0. A numerical study is presented in .The main goal in this work is to study the asymptotic behavior of the extremal parameters and solutionsof (1). More precisely, we show that the limit lim ε→0 λ ∗ ε exists (at least for a sequence ε i → 0), andwe identify it as the extremal parameter of some limit problem. Similarly, we prove that the extremalsolutions u ∗ ε converge in some sense, to the extremal solution of a limit problem.1.2. Definitions and main resultsWhen dealing with the nonlinear problem (1) it is important to know the asymptotic behavior ofsolutions of a linear equation with the same boundary condition as in (1), namely⎧−∆u ε + u ε = h in Ω,⎪⎨u ε = 0 onΓ1 ε,(3)∂u ⎪⎩ ε∂ν = 0 onΓ 2 ε, where h ∈ L 2 (Ω).It turns out that a convenient class of linear problems to consider, is⎧⎨ −∆u + u + σu = h in Ω,∂u⎩∂ν + σu = 0 on∂Ω, (4)where h ∈ L 2 (Ω) andσ belongs to a certain class of Borel measures. The main reference that we usehere for the linear problem (4) and questions on the asymptotic behavior of their solutions is Buttazzo etal. . Other references are [10,8,9].Definition 1.1.(a) M denotes the collection of Borel measures on R n with values into [0, ∞] thatvanishonBorelsets of capacity zero and have support in Ω.(b) For σ ∈Mwe set H σ = H 1 (Ω) ∩ L 2 (Ω, σ) which is a Hilbert space with the inner product∫∫〈u, ϕ〉 = ∇u∇ϕ + uϕ dx + ũ ˜ϕ dσ,ΩΩwhere ũ and ˜ϕ are quasi-continuous representatives of u and ϕ.(c) We say that u is an H 1 -solution of (4) if u ∈ H σ and∫Ω∫∫∇u∇ϕ + uϕ dx + ũ ˜ϕ dσ = hϕ dx for all ϕ ∈ H σ .ΩΩ

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 281Remarks.1) We note here that the integrals with respect to the measure σ are well defined for u, ϕ ∈ H σbecause σ vanishes on sets of capacity zero, and quasi-continuous representatives of an element inH 1 (Ω) agree up to sets of capacity zero (see ). From now on we drop the “˜”inũ, ˜ϕ and alwaysuse quasi-continuous representatives in integrals with respect to a measure σ ∈M.2) Problem (4) has a unique solution, which is also the minimizer of∫∫∫|∇u| 2 + u 2 dx + u 2 dσ − 2 hu dx.ΩΩΩA trivial case which can occur is when for all Borel sets B, σ(B) = ∞ if B ∩ Ω has positivecapacity, and σ(B) = 0 otherwise. Then H σ = {0}, and in this case 0 is the solution of (4) forany h.3) A mixed boundary condition as in (3) can be obtained by taking{ ∞ if B ∩ Γεσ ε (B) =1 has positive capacity,0 otherwisefor all Borel sets B.4) If supp(σ) ⊂ ∂Ω, then (4) can also be rewritten in the form⎧⎨ −∆u + u = h in Ω,∂u⎩∂ν + σu = 0 on ∂Ω.5) Here is an example in which the measures have support inside Ω. Consider a union of disjoint ballsT = ⋃ i B i ,andlet ˜Ω = Ω \T (this is usually called a perforated domain, and the balls are usuallytaken in a periodic arrangement). Taking σ(B) = ∞ if B ∩ (T ∪ ∂Ω) has positive capacity, andσ(B) = 0 otherwise, (4) can be written as{ −∆u + u = h in ˜Ω,u = 0on ∂ ˜Ω.We consider the following notion of convergence for measures in M.Definition 1.2. If (σ i ) ⊂Mis a sequence of measures we write σ ih ∈ L 2 (Ω), the solutions u i of⎧⎨ −∆u i + u i + σ i u i = h in Ω,∂u⎩ i∂ν + σ iu i = 0 on ∂ΩB ⇀σ∞ where σ ∞ ∈Mif for all(5)satisfy u i ⇀u ∞ in H 1 (Ω) weakly as i →∞,whereu ∞ is the solution of⎧⎨ −∆u ∞ + u ∞ + σ ∞ u ∞ = h in Ω,∂u⎩ ∞∂ν + σ ∞u ∞ = 0 on ∂Ω.

282 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsObserve that we formulate this definition for the operator −∆+I instead of −∆, which would be morenatural for the nonlinear problem (1). The advantage of this formulation is that the solution u i of (5) isbounded in H 1 (Ω) without any assumption on σ i or h.As an example, in the case in which Ω is the unit disk in R 2 , ∂Ω is subdivided in segments of length εand the boundary condition is zero Dirichlet and zero Neumann on alternate segments, the limit boundarycondition in the sense of Definition 1.2 is a zero Dirichlet condition. This is shown in Example 1 ofSection 2.2. That section contains also some other examples.The following compactness theorem is a consequence of the results in .Theorem 1.3. If (σ i ) ⊂Mis a sequence, then there is a subsequence (σ ij ) and σ ∞ ∈Msuch thatB ⇀σ∞ . Moreover, if supp(σ i ) ⊂ ∂Ω for all i, thensupp(σ ∞ ) ⊂ ∂Ω.σ ijNext we consider the nonlinear problem⎧⎨ −∆u + σu = λf(u) in Ω,∂u⎩∂ν + σu = 0 on ∂Ω, (6)where σ ∈M, σ ≢ 0. Recall that f(u) > 0 and we are interested in λ>0. If σ ≡ 0 then (6) has nosolution for λ>0. On the other hand, the hypothesis σ ≢ 0 implies that for any ϕ ∈ L 2 (Ω) there is aunique solution ζ ∈ H σ of⎧⎨ −∆ζ + σζ = ϕ in Ω,∂ζ⎩∂ν + σζ = 0 on∂Ω.We use the notationζ = T σ (ϕ) (7)and this defines a bounded linear operator T σ : L 2 (Ω) → H σ .Definition∫1.4. Let σ ∈M with σ ≢ 0. We say that u ∈ L 1 (Ω) is a weak solution of (6) ifΩ f(u)χ< ∞ where χ = T σ(1), and for all ϕ ∈ C0 ∞(Ω)wehave∫∫uϕ dx = λ f(u)T σ (ϕ)dx.ΩΩRemark. In the case of the zero Dirichlet boundary condition, this is the same notion of weak solutionintroduced by Brezis et al. . In this case, the test functions ζ = T σ (ϕ) belong to C 2 (Ω) and vanishon the boundary in the usual sense. But for a general σ ∈Mit is hard to describe the precise regularityof ζ.Proposition 1.5. Assume σ ∈Mis not identically zero and that H σ ≠ {0}. Then there exits λ ∗ ∈ (0, ∞)such that for 0

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 283Important notation. In order to state the main results, for a given σ ∈Mwith σ ≢ 0andH σ ≠ {0},we let λ ∗ (σ) denote the corresponding extremal parameter of (6). If additionally supp(σ) ⊂ ∂Ω we letu ∗ (σ) be the extremal solution of (6). Note that if σ ≡ 0 then (6) has no solution for any λ>0, so weuse the convention λ ∗ (σ) = 0. On the other hand, if H σ = {0} we use the convention λ ∗ (σ) = ∞.Theorem 1.6. If (σ i ) ⊂Mis a sequence such that σ iB ⇀σ∞ thenlimiλ ∗ (σ i ) = λ ∗ (σ ∞ ). (8)In particular we find lim ε→0 λ ∗ ε in the example where Ω ⊂ R2 , ∂Ω is subdivided in segments oflength ε, with zero Dirichlet and Neumann conditions on alternate segments. The result states thatlim ε→0 λ ∗ ε is the extremal parameter for the same nonlinear equation but with zero Dirichlet boundarycondition.On the asymptotic behavior of the extremal solution, we have the following result:Theorem 1.7. Let (σ i ) ⊂Mbe sequence such that supp(σ i ) ⊂ ∂Ω for all i and σ iσ ∞ ≢ 0. ThenB ⇀σ∞ ,whereu ∗ (σ i ) → u ∗ (σ ∞ ), as i →∞, (9)in L p (Ω) for 1 p

284 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsLet g ∈ C ∞ (R n ), g>0inΩ, g = 0inR n \ Ω and let v i denote the solution of{ −Lε iv i + v i + σ i v i = g in R n ,v i ∈ H 1 (R n ).(10)The variational formulation of (10) is∫∫(ε i 1 R n \Ω + 1 Ω )∇v i ∇ϕ + v i ϕ dx +R nΩ∫v i ϕ dσ i = gϕ dxR n (11)for all ϕ ∈ H 1 (R n ) ∩ L 2 (R n , σ i ).By Theorems 4.1 and 5.1 of  we have that there is a subsequence σ ij and a measure σ ∞ ∈Msupported in Ω such that v ij → v ∞ in L 2 (R n )wherev ∞ = g = 0inR n \ Ω and v ∞ | Ω= v 0 where v 0is the solution of⎧⎨ −∆v 0 + v 0 + σ ∞ v 0 = g in Ω,∂v⎩ 0∂ν + σ ∞v 0 = 0 on∂Ω.We mention here that if supp(σ i ) ⊂ ∂Ω for all i, then by [6, Lemma 6.2] we have supp(σ ∞ ) ⊂ ∂Ω.Let h ∈ L 2 (Ω), let u i denote the solution of⎧⎨ −∆u i + u i + σ i u i = h in Ω,∂u⎩ i∂ν + σu i = 0 on ∂Ω(12)(13)and u ∞ denote the solution of⎧⎨ −∆u ∞ + u ∞ + σ ∞ u ∞ = h in Ω,∂u⎩ ∞∂ν + σ ∞u ∞ = 0 on ∂Ω.(14)Note that (13) implies that u i is bounded in H 1 (Ω), so that for a further subsequence we can assume thatu i ⇀uin H 1 (Ω) weakly. From now on we will just use the index i for all subsequences. To conclude,we only need to show that u = u ∞ where u ∞ is the solution of (14). We start with the case h ∈ L ∞ (Ω).The general case can then be obtained by a density argument.Let ζ ∈ C ∞ 0 (Rn )andletususeζv i as a test function in the variational formulation of (13). Note thatv i is bounded, so that ζv i ∈ H 1 (Ω) and also note that ζv i ∈ L 2 (Ω, σ i ). Thus we obtain∫Ω∫∫ζ∇u i ∇v i + v i ∇u i ∇ζ + u i v i ζ dx + u i v i ζ dσ i = hv i ζ dx. (15)ΩΩNow we need to extend u i ∈ H 1 (Ω) toR n . We denote by E : H 1 (Ω) → H 1 (R n ) a linear boundedextension operator, with the property that ‖Ew‖ L ∞ (R n ) C‖w‖ L ∞ (Ω). Setnowū i = Eu i .Wewantto use ϕ =ū i ζ in (11). Remark that since we assume h ∈ L ∞ (Ω) wehavethatu i ∈ L ∞ (Ω) andso

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 285ū i ∈ L ∞ (R n ). Therefore ū i ζ ∈ H 1 (R n ) and we also have ū i ζ ∈ L 2 (R n , σ i ). Hence we obtain∫∫∫∫ε i ∇v i ∇(ū i ζ)dx + ζ∇v i ∇ū i +ū i ∇v i ∇ζ dx + v i ū i ζ dx + v i ū i ζ dσ iR n \ΩΩR n Ω∫= gu i ζ dx. (16)ΩWe now subtract (15) from (16):∫∫∫∫ε i ∇v i ∇(ū i ζ)dx + (u i ∇v i − v i ∇u i )∇ζ dx + v i ū i ζ dx − v i u i ζ dxR n \ΩΩR n Ω∫= (gū i − hv i )ζ dx. (17)ΩWe want now to pass to the limit as i →∞. For this observe that from (11) (with ϕ = v i )wefind∫R n (ε i 1 R n \Ω + 1 Ω )|∇v i | 2 + v 2 i dx + ∫Ω∫vi 2 dσ i =Ωgv i dx. (18)This shows that v i | Ω is bounded in H 1 (Ω) and therefore converges weakly in H 1 (Ω) tov 0 , which is thesolution of (12). But also from (18) we find that∫ε i |∇v i | 2 dx CR n \Ωwith C independent of i. We use this to estimate the first term on the left-hand side of (17):ε i∫R n \Ω∫) 1/2 ( ∫1/2∇v i ∇(ū i ζ)dx ε 1/2i(ε i |∇v i | 2 dx∣ ∇(ū i ζ) ∣ dx) 2 → 0R n \ΩR n \Ωas i →∞. So, taking the limit as i →∞in (17) we arrive at∫Ω∫(u∇v 0 − v 0 ∇u)∇ζ dx =Ω(gu − hv 0 )ζ dx. (19)We note that (19) is also satisfied if we replace u by u ∞ . This can be seen by using v 0 ζ in the variationalformulation of (14), then taking ϕ = u ∞ ζ in the variational formulation of (12) and subtracting. Hence,if we set ũ = u − u ∞ , we obtain∫Ω(ũ∇v0− v 0 ∇ũ ) ∫∇ζ dx =Ωgũζ dx (20)for all ζ ∈ C ∞ 0 (Rn ) and hence for all ζ ∈ C ∞ (Ω). Remark that u i is bounded in L ∞ (Ω) and thereforeũ ∈ L ∞ (Ω). Also v 0 ∈ L ∞ (Ω), so (20) is valid for all ζ ∈ H 1 (Ω). We take ζ = ũ in (20) and obtain∫Ω12 ∇v 0∇ ( ũ ) 2 ∣ − v0 ∇ũ ∣ ∫2 dx = gũ 2 dx. (21)Ω

286 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsBut taking ϕ = ũ 2 in the variational formulation of (12) we find∫Ω∇v 0 ∇ ( ũ ) ∫∫2 + v0 ũ 2 dx + v 0 ũ 2 dσ ∞ = gũ 2 dx. (22)ΩΩCombining (21) and (22) we obtain∫gũ 2 ∣+ 2v 0 ∇ũ ∣ ∫2 + v 0 ũ 2 dx + v 0 ũ 2 dσ ∞ = 0.ΩΩSince g>0inΩ we conclude that ũ = 0, and therefore u = u ∞ .2.2. Some examplesThere are many examples in the literature.Example 1. This example includes the one mentioned in the introduction, in which Ω is the unit diskin R 2 , ∂Ω is divided in segments of length ε and a zero Dirichlet and Neumann condition is applied onalternate segments.More generally, suppose that Γ1 ε, Γ 2 ε is a family of partitions of ∂Ω that satisfies the following conditions:limε→0sup dist ( x, Γ ε )1 = 0 (23)x∈∂Ω(with this we want to capture the notion that the partition becomes finer as ε → 0), and⎧⎪⎨ there exist ρ 0 > 0, ν 0 > 0 such that for all y ∈ Γ1 ε and all 0

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 287Proof of (25). Fix some h ∈ L ∞ (Ω)andletu ε be the solution of⎧−∆u ε + u ε = h in Ω,⎪⎨u ε = 0 onΓ1 ε,(26)∂u ⎪⎩ ε∂ν = 0 onΓ 2 ε. Since the partitions Γ ε 1 , Γ ε 2 satisfy (24) with constants independent of ε, by Theorem 3.4 u ε is boundedin C α (Ω) forsomeα ∈ (0, 1). Hence, by taking a subsequence we can assume that u ε → u uniformlyin Ω. But then, by (23) u| ∂Ω = 0. Now let ζ ∈ C 2 (Ω) with ζ| ∂Ω = 0. By (26) we have∫Ω∫∂ζu ε (−∆ζ + ζ)dx + u ε∂Ω ∂ν∫Ωds = hζ dxand taking the limit as ε → 0wefindthatu is the solution of{ −∆u + u = h in Ω,u = 0on∂Ω.Example 2. There are some examples by Cioranescu and Murat , where the measures in questionhave support inside Ω. We refer to their article for the detailed description of the results.Example 3. This example is a consequence of the results of Damlamian for the Neumann sieve .We mention it in connection with Example 1, to show what happens if the local proportion Γ1 ε (the partof ∂Ω where we set u ε = 0) goes to zero at a certain speed.More concretely, suppose that a portion Σ of the boundary ∂Ω is contained in the hyperplane {x n = 0}(we use the standard notation x = (x ′ , x n ) ∈ R n with x ′ ∈ R n−1 and x n ∈ R), and that Ω ⊂ R n + ={x n > 0}.Let {Γ1 ε, Γ 2 ε } denote a family of partitions of ∂Ω such that:1) Γ1 ε ∩ Σ is a periodic arrangement with period εY , Y = (0, 1)n−1 , of sets Oε. i Each Oε i is assumedto be, up to a translation, equal to b ε O,whereO⊂R n−1 is the reference set, and b ε > 0isthe“size” of Oε i , to be defined later as a function of ε.2) ∂Ω \ Σ ⊂ Γ1 ε.Let h ∈ L 2 (Ω)andletu ε be the solution of⎧−∆u ε + u ε = h in Ω,⎪⎨u ε = 0 onΓ1 ε,∂u ⎪⎩ ε∂ν = 0 onΓ 2 ε.Claim. Assume that O (the reference set) is a bounded, open, smooth subset of R n−1 , n 3, andb ε = ε (n−1)/(n−2) .Thenu ε ⇀u in H 1 (Ω) weakly, (27)

288 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionswhere u is the solution of⎧−∆u + u = h in Ω,⎪⎨∂u⎪⎩∂ν + c u = 0 on Σ,2u = 0 on ∂Ω \ Σand c>0 is the capacity in R n of O×{0}. We highlight here the boundary condition on Σ:∂u∂ν + c 2 u = 0onΣ.This can be rephrased in terms of measures as well.From the work in  one can also see that if b ε = o(ε (n−1)/(n−2) ) in the case n 3, then the limitboundary condition on Σ is a zero Neumann condition.Sketch of the proof of (27). Defineũ ε( x ′ , x n) ={ uε (x ′ , x n ) ifx n > 0,−u ε (x ′ , −x n ) if x n < 0.By [11, Theorem 1] ũ ε ⇀ ũ in V ,whereV is the Hilbert space H 1 (Ω) × H 1 (Ω − ), Ω − is the reflectionof Ω across {x n = 0}, and ũ solves⎧−∆ũ + ũ =⎪⎨ũ = 0in Ω ∪ Ω − ,on ∂Ω ∪ ( ∂Ω −) \ Σ,⎪⎩∂ũ∂ν − = ∂ũ∂ν + = − c [ũ]4on Σ.Here ∂/∂ν − and ∂/∂ν + are the normal derivatives of ũ coming from Ω − and Ω, respectively (recall thatν points to the outside of Ω, so∂/∂ν = −∂/∂x n ), and [ũ] = ũ + − ũ − ; ũ + , ũ − being the values of ũon Σ when coming from Ω and Ω − , respectively.But ũ is odd across Σ, so the jump condition in (28) may be written as(28)∂ũ∂ν + c 2ũ+ = 0onΣ.3. PreliminariesIn this section we collect a number of preliminary results that are needed later. We denote by σ afixedelement in M with σ ≢ 0.Recall that we defined H σ = H 1 (Ω) ∩ L 2 (Ω, σ) which is a Hilbert space with the inner product∫〈u, v〉 σ =Ω∫∇u∇v + uv dx + uv dσ.Ω

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 289The assumption σ ≢ 0 implies that there is a constant C>0 (depending on σ and Ω) such that for allϕ ∈ H σ∫Ω( ∫∫ )ϕ 2 dx C |∇ϕ| 2 dx + ϕ 2 dσΩΩor equivalently, that the first eigenvalue of −∆ + σ| Ω , with the generalized Robin boundary condition∂ϕ/∂ν + σϕ = 0on∂Ω, is positive:λ 1 (σ) =∫Ωinf|∇ϕ|2 dx + ∫ Ω ϕ2 dσ∫ϕ∈H σ Ω ϕ2 dx> 0. (29)Note that it can happen that H σ = {0}. In this case we adopt the convention λ 1 (σ) = ∞.If σ ∈Mand λ 1 (σ) < ∞, then the infimum in (29) is attained at some nonnegative, nonzero functionϕ 1 ∈ H σ which we call the first eigenfunction associated to σ. It satisfies the equation⎧⎨ −∆ϕ 1 + σϕ 1 = λ 1 (σ)ϕ 1 in Ω,∂ϕ⎩ 1∂ν + σϕ 1 = 0 on∂Ω.We remark here that in many elliptic estimates in this and later sections, we will say that the constantsdepend on σ only through λ 1 (σ), meaning that these constants remain bounded as long as λ 1 (σ) isbounded away from zero.3.1. Some elliptic estimatesThe first result we mention here is an L ∞ bound. Its proof is standard, and follows that of Lemma 7.3of Hartman and Stampacchia .Proposition 3.1. Let p>n/2. Then there exists a constant C>0 depending only on Ω, n, p and λ 1 (σ)such that for any solution u of⎧⎨ −∆u + σu = h in Ω,∂u⎩∂ν + σu = 0 on ∂Ωwith h ∈ L p (Ω) we have‖u‖ ∞ C‖h‖ p .The next result is also important (see ).Lemma 3.2. Assume that σ ∈Mhas support on ∂Ω. Letχ be the H 1 -solution of⎧⎨ −∆χ = 1 in Ω,∂χ⎩∂ν + σχ = 0 on ∂Ω.

290 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsSuppose that ζ is the H 1 -solution of⎧⎨ −∆ζ = ϕ in Ω,∂ζ⎩∂ν + σζ = 0 on ∂Ω,where ϕ ∈ L p (Ω), p>n. Then there exists C such thatζ∥χ∥ C‖ϕ‖ p . (30)∞The constant C depends on Ω, n, p and λ 1 (σ).Remark 3.3. We mention that the assumption supp(σ) ⊂ ∂Ω is not absolutely necessary. It is enoughthat the support of σ is contained in ∂Ω ∪ K where K is a compact smooth n − 1 dimensional manifoldcontained in Ω.Another observation is that in  the result is stated for a mixed boundary condition, but the proofgiven there works also for a measure σ ∈Mwith supp(σ) ⊂ ∂Ω.Under some extra assumptions on σ it is possible to establish the Hölder continuity of the solutions(this is an adaptation of a result of Stampacchia ).Theorem 3.4. Suppose u is a solution of⎧−∆u = h in Ω,⎪⎨u = 0 on Γ 1 ,⎪⎩ ∂u∂ν + σu = g on Γ 2,where Γ 1 , Γ 2 is a partition of ∂Ω, h ∈ L p (Ω), p>n/2, and σ, g ∈ L q (Γ 2 ), q>n− 1. We assume thefollowing “regularity” condition:there exists ρ 0 > 0, ν 0 > 0 such that for all y ∈ Γ 1 and all 0

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 291Definition 3.5.(a) Let χ = T σ (1) (T σ wasdefinedin(7)).(b) We introduce L 1 χ = L 1 (Ω, χ dx) and‖h‖ L 1 χ= ∫ Ω |h|χ.(c) Let h ∈ L 1 χ. We say that u ∈ L 1 (Ω) is a weak solution of (32) if∫Ω∫uϕ dx =Ωfor any ϕ ∈ C ∞ 0 (Ω).hT σ (ϕ)dx (33)Remarks.1) The functions ζ = T σ (ϕ) ∈ H σ as in the previous definition play the role of the test functionsζ ∈ C 2 (Ω) with ζ| ∂Ω = 0 in the case of a Dirichlet boundary condition (see ).2) Observe also that any H 1 -solution is a weak solution.3) Note that ∫ Ω |hT σ(ϕ)| dx 0depending only Ω, n, p and λ 1 (σ) such that if u is the weak solution of (32) then‖u‖ p C‖h‖ L 1 χ.Proof. We use a duality argument. Let p ′ denote the conjugate exponent of p (that is 1/p + 1/p ′ = 1)and let ϕ ∈ C ∞ 0 (Ω)andζ = T σ(ϕ). Then from (33) we find∫Ω∫uϕ dx =∥ ∥∥∥ ζhζ dx ‖h‖ ∥ L 1 χΩχ∥ ∞ C‖h‖ L 1 χ‖ϕ‖ p ′,where the last inequality is a consequence of (30) (note that since 1 pn). Remark. Again, we can relax the assumption on the support of σ as in Remark 3.3.Definition 3.8. Let h ∈ L 1 χ . We say that u ∈ L1 (Ω) is a weak supersolution of (32), which we denoteby⎧⎨ −∆u + σu h in Ω,∂u⎩∂ν + σu 0 on∂Ω

292 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsif for any ϕ ∈ C ∞ 0 (Ω) such that T σ(ϕ) 0wehave∫Ω∫uϕ dx ΩhT σ (ϕ)dx.The two following versions of the strong maximum principle for −∆ with Robin boundary conditionare consequences of Lemma 3.2 (see ).Theorem 3.9. Assume supp(σ) ⊂ ∂Ω. Then there exists c>0 depending only on Ω and λ 1 (σ) such thatif h ∈ L 1 χ and u is a solution of (32) then( ∫u(x) c hχ)χ(x) a.e. in Ω.ΩLemma 3.10. Assume supp(σ) ⊂ ∂Ω and suppose that u is a supersolution of (32) with h = 0. Theneither u ≡ 0 or there exists c>0 such thatu cχ a.e. in Ω.Finally, an important tool is the following result (see the case of zero Dirichlet condition in [4,3]).Lemma 3.11 (Kato’s inequality). Let h ∈ L 1 σ and u ∈ L 1 (Ω) a weak solution of (32). Let Φ : R → R bea C 2 concave function with Φ ′ ∈ L ∞ and Φ(0) = 0. Then⎧⎨ −∆Φ(u) + σΦ(u) Φ ′ (u)h in Ω,∂Φ(u)⎩ + σΦ(u) 0 on ∂Ω.∂νFor completeness we give a proof in the appendix.3.3. The nonlinear problemIn this section we consider the nonlinear problem⎧⎨ −∆u + σu = λf(u) in Ω,∂u⎩∂ν + σu = 0 on ∂Ω. (35)Definition 3.12. We say that u ∈ L 1 (Ω) is a weak solution of (35) if f(u) ∈ L 1 χ and∫∫uϕ dx = λ f(u) T σ (ϕ)dxΩΩfor any ϕ ∈ C0 ∞(Ω).We also say that U ∈ L 1 (Ω) is a weak supersolution of (35) if f(U) ∈ L 1 χ and∫Ω∫Uϕdx Ωf(U) T σ (ϕ)dxfor any ϕ ∈ C ∞ 0 (Ω) such that T σ(ϕ) 0.

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 293Lemma 3.13. Suppose that U is a weak supersolution of (35). Then(35) has a minimal solution0 u U.The proof is analog to the case of zero Dirichlet boundary condition. See  for example.The following theorem summarizes some of the properties of (35).Theorem 3.14. Let σ ∈Mwith σ ≢ 0 and suppose that H σ ≠ {0}. Then:(i) There exists 0

294 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsRemarks.1) Most of these results are adaptations of the analog statements for the Dirichlet boundary conditionusing mainly Lemmas 3.13 and 3.14, and we refer to the literature [4,2,5]. The proof of (v), onthe other hand, requires a new result: a strong maximum principle with Robin boundary conditionwhich is given in Lemma 3.10. With it is possible to adapt the argument given by Martel  forthe case of zero Dirichlet boundary condition.2) If σ is not supported on ∂Ω, but on ∂Ω ∪ K with K a compact smooth n − 1 dimensional manifoldcontained in Ω, then the conclusions of the theorem still hold.3) In the general case we can always consider the monotone limit u ∗ = lim λ↗λ ∗ u λ . It can be shownto exist pointwise, and it satisfies∫∫u ∗ ϕ 1 dx

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 295BBy assumption of σ i ⇀σ∞ and since λ 1 (σ ∞ ) = ∞ we have ζ i ⇀ 0inH 1 (Ω) weakly.Now we multiply (37) by ζ i and integrate by parts, multiply (38) by ϕ i and integrate by parts, and takethe difference to obtain∫∫ζ i ϕ i dx = hϕ i − λ i ϕ i ζ i dx.ΩΩBut ζ i ⇀ 0andϕ i ⇀ϕin H 1 (Ω) weakly, so∫Ωhϕ dx = 0.Since h ∈ L 2 (Ω) was arbitrary we conclude that ϕ = 0, but this is in contradiction with ‖ϕ‖ L 2 = 1.Step 2. If λ ∞ < ∞ then there exists C

296 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsObserve that if σ ∞ ≢ 0thenζ is uniquely defined, and otherwise ζ is defined up to constant. Let ζ idenote the solution ofClaim.⎧⎨ −∆ζ i + ζ i + σ i ζ i = h + ζ in Ω,∂ζ⎩ i∂ν + σ iζ i = 0on ∂Ω.ζ i ⇀ζ in H 1 (Ω) weakly. (41)Proof of Lemma 3.15 completed. Multiplying (40) by ϕ i , integrating by parts and using (37) we find∫Ω∫λ i ϕ i ζ i + ζ i ϕ i =Ωhϕ i + ζϕ iso that by letting i →∞we have∫ ∫λ ϕζ = hϕ. (42)Ω Ω(40)In the case σ ∞ ≡ 0, since we could replace ζ by ζ + c in (42), we conclude that λ = 0 = λ 1 (σ ∞ ).In the case σ ∞ ≢ 0, from (42) we deduce that ϕ satisfies⎧⎨ −∆ϕ + σ ∞ = λϕ in Ω,∂ϕ⎩∂ν + σ ∞ϕ = 0 on∂Ω.(43)Since ϕ ≢ 0, ϕ 0, (43) implies that λ = λ 1 (σ ∞ ).Proof of (41). By definition of σ iB ⇀σ∞ we have ζ i ⇀ ˜ζ in H 1 (Ω) weakly, where ˜ζ is the solution of⎧⎪⎨ −∆˜ζ + ˜ζ + σ ˜ζ ∞ = h + ζ in Ω,∂˜ζ ⎪⎩∂ν + σ ˜ζ ∞ = 0on ∂Ω.But −∆ζ + σ ∞ ζ = h so that⎧⎪⎨ −∆ (˜ζ − ζ ) + (˜ζ − ζ ) + σ ∞(˜ζ − ζ ) = 0inΩ,( ) ∂ ⎪⎩∂ν + σ ∞(˜ζ − ζ ) = 0 on∂Ωso that ζ = ˜ζ.

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 297BLemma 3.16. Assume σ i ⇀σ∞ where σ ∞ ≢ 0. By Lemma 3.15 we have that λ 1 (σ i ) is bounded awayfrom zero for i large. Let ϕ ∈ L 2 (Ω) and ζ i be the solution of⎧⎨ −∆ζ i + σ i ζ i = ϕ in Ω,∂ζ⎩ i∂ν + σ iζ i = 0 on Ω.Then ζ i ⇀ζ ∞ in H 1 (Ω) weakly where ζ ∞ is the solution of(44)⎧⎨ −∆ζ ∞ + σ ∞ ζ ∞ = ϕ in Ω,∂ζ⎩ ∞∂ν + σ ∞ζ ∞ = 0 on Ω.(45)Proof. Since λ 1 (σ i ) is bounded away from zero, we have that ‖ζ i ‖ H 1 C for some C independent of i,and therefore up to subsequence ζ i ⇀ζin H 1 (Ω) weakly. We let v i denote the solution of⎧⎨ −∆v i + v i + σ i v i = ϕ + ζ in Ω,∂v⎩ i∂ν + σ iv i = 0on Ω(46)so that by definition v i ⇀vin H 1 (Ω) weakly to v ∞ which is the solution of⎧⎨ −∆v ∞ + v ∞ + σ ∞ v ∞ = ϕ + ζ in Ω,∂v⎩ ∞∂ν + σ ∞v ∞ = 0 on Ω.(47)Then by (44) and (46) we have‖v i − ζ i ‖ H 1 ‖ζ − ζ i ‖ L 2 → 0and this implies that v ∞ = ζ. But then, by (47) we see that ζ satisfies (45) and by uniqueness of thesolution of this problem we have ζ = ζ ∞ . 4. Convergence of the extremal parameterThroughout this section (σ i ) i is a sequence in M such that σ iλ ∗ i = λ∗ (σ i ), λ ∗ ∞ = λ∗ (σ ∞ ).We divide the proof of Theorem 1.6 in two steps.BStep 1. If σ i ⇀σ∞ ,thenB ⇀ σ∞ , and we use the notationlim sup λ ∗ i λ∗ ∞ .i

298 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsProof. If λ ∗ ∞ = ∞ there is nothing to prove, so we assume that λ∗ ∞ < ∞. Suppose that the conclusionis not true, and take a subsequence (which we denote the same) such that λ ∗ i → λ with λ∗ ∞

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 299But subtracting (48) from (50) we get:‖w i − v i ‖ H 1 (Ω) C ∥∥ λ ′ f(v) − λ ′ f(v i ) + v − v i∥ ∥L 2 (Ω) → 0.Hence we must have v = w.Step 2.lim inf λ ∗ i λ ∗ ∞.iProof. If the conclusion is not true, then there exists a subsequence (denoted the same) such that λ ∗ i →λ

300 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsthen we have shown that w i is a supersolution for the problem⎧⎨ −∆u + σ i u = λ ′′ f(u) in Ω,∂u⎩∂ν + σ iu = 0 on ∂Ωand this contradicts the fact that λ ∗ iis the maximal parameter for this nonlinear problem.Proof of (55). Subtracting (52) from (53) and using Proposition 3.1 we find that‖w i − v i ‖ ∞ C ∥∥ λ ′ f ( u ′) + u ′ − λ ′ f(v i ) − v i∥ ∥p ,wherewefixsomen/2

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 301where 1 p

302 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsBy Lemma 3.16 we have that ζ i ⇀ζin H 1 (Ω) weakly, where ζ is the solution of⎧⎨ −∆ζ = ϕ in Ω,∂ζ⎩∂ν + σ ∞ζ = 0 on∂Ω.Note that since ϕ is smooth, we have that ζ i is bounded in Cloc k (Ω) foranyk 0, and therefore ζ i → ζin Cloc k (Ω)foranyk 0. In particular we have a.e. convergence. Taking ζ i as a test function in the weakformulation of⎧⎨ −∆u ∗ i = λ∗ i f ( u ∗ )i in Ω,⎩ ∂u ∗ i∂ν + σ iu ∗ i = 0 on∂Ωwe find∫Ω∫u ∗ i ϕ dx = λ ∗ iΩf ( u ∗ i)ζi dx.By passing to the limit as i →∞and using Fatou’s lemma on the right-hand side we find∫Ωuϕ dx λ ∗ ∞∫Ωf(u)ζ dx.This shows that u is a weak supersolution of⎧⎨ −∆u = λ ∗ ∞f(u) in Ω,∂u⎩∂ν + σ ∞u = 0 on ∂Ω.and this finishes the proof of (9) in Theo-By Theorem 3.14 property (v), we conclude that u = u ∗ ∞rem 1.7. 5.2. Asymptotic behavior of sup Ω u ∗ (λ i )In this section we prove the second part of Theorem 1.7, which we recall now: if u ∗ ∞ is unboundedthen∥∥ ∞→∞∥u ∗ iand if u ∗ ∞ ∈ L ∞ (Ω)thenlim sup ∥∥ u ∗ (σ i ) ∥ ∥ ∞< ∞.Step 1. If u ∗ is unbounded then∥ u ∗ ∥i ∞→∞.

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 303Proof. This is a consequence of the fact thatu ∗ i → u∗ ∞ in Lp (Ω), 1 p< nn − 1 .Step 2. If u ∗ ∞ ∈ L ∞ (Ω) thenlim sup ∥∥ u ∗ (σ i ) ∥ ∥ ∞< ∞.Proof. Suppose not and consider a subsequence (denoted the same) such that sup Ω u ∗ i ↗∞.Wefixnow M = C 1 + 2 < ∞, whereC 1 is to be chosen later. Now, for each fixed i because of property (vii)in Theorem 3.14 we can select 0

304 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionswhere C 1 depends on λ ∗ ∞ , u∗ ∞ , Ω, n and λ 1(σ i ), which is bounded away from zero. At this point wemake the choice of C 1 .Recall that we assume u ∗ ∞ ∈ L ∞ (Ω), hence by Lemma 3.16 we have v i ⇀u ∗ ∞ in H 1 (Ω) weakly. Butsubtracting (64) from (60) and using Proposition 3.1 we havesup u i − v i C ∥ ( λ i f(u i ) − λ ∗ ∞f ( u ∗ + ∥ ∥p∞)),Ωwherewefixsomen/2

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 305Now, multiplying (65) by ζ i and integrating by parts we get∫∫∫|∇ζ i | 2 + ζi 2 dσ i = ϕζ i .ΩΩΩSince ζ i ⇀ζin H 1 (Ω) weakly, this equality shows that∫∫∫ ∫|∇ζ i | 2 + ζi 2 dσ i → |∇ζ| 2 + ζ 2 dσ ∞ .ΩΩΩΩTaking i →∞in (68) and using Fatou’s lemma on the right-hand side, we obtain (67) for ζ in a subsetof H σ∞ , namely the ones that are solutions of (66) for some ϕ ∈ C0 ∞ (Ω). But this subset is dense inH σ∞ and (67) follows.By Theorem 3.14 property (i) we must have ˜λ λ ∗ ∞ , and by property (ix) of the same theorem ũ isthe minimal solution of (63). AppendixProof of Lemma 3.11. Recall that we assume that u is a weak solution of⎧⎨ −∆u + σu = h in Ω,∂u⎩∂ν + σu = 0 on∂Ω,where σ ∈Mand h ∈ L 1 χ.Form>0leth m = h if |h| m, h m = −m if hm,andletu m denote the H 1 -solution of⎧⎨ −∆u m + σu m = h m in Ω,∂u⎩ m∂ν + σu (69)m = 0 on ∂Ω.Note that u m → u in L 1 (Ω). Let ϕ ∈ C0 ∞ (Ω) and suppose that the solution ζ of⎧⎨ −∆ζ + σζ = ϕ in Ω,∂ζ⎩∂ν + σζ = 0 on∂Ω (70)is nonnegative.Note that Φ ′ (u m )ζ ∈ H σ because Φ ′ ∈ L ∞ , ζ ∈ H σ and ∇(Φ ′ (u m )ζ) ∈ L 2 (Ω). Using Φ ′ (u m )ζ as atest function in (69) we find that∫(∇u m Φ ′′ (u m )∇u m ζ + Φ ′ (u m )∇ζ ) ∫∫dx + Φ ′ (u m )u m ζ dσ = h m Φ ′ (u m )ζ dx.ΩΩBut Φ ′′ 0 because Φ is concave, and Φ ′ (u)u Φ(u) (this follows from the concavity of Φ andΦ(0) = 0). Hence∫∇ ( Φ(u m ) ) ∫∫∇ζ dx + Φ(u m )ζ dσ h m Φ ′ (u m )ζ dx. (71)ΩΩΩΩ

306 J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditionsNote that Φ(u m ) ∈ H σ because Φ(u) ‖Φ ′ ‖ ∞ |u| ∈L 2 (Ω, σ). Using Φ(u m ) in (70) we obtain∫Ω∇ ( Φ(u m ) ) ∫∫∇ζ dx + Φ(u m )ζ dσ = Φ(u m )ϕ dx. (72)ΩΩCombining (71) and (72) we get∫∫Φ(u m )ϕ dx h m Φ ′ (u m )ζ dx.ΩΩNow we let m →∞:∫∣ Φ(u m ) − Φ(u) ∣∣ ∥|ϕ| dx ‖ϕ‖ ∞ Φ ′∥ ∫∥ ∞ΩΩ|u m − u| dx → 0and∫∫h m Φ ′ (u m )ζ dx → hΦ ′ (u)ζ dxΩΩsince we have convergence a.e. (at least for a subsequence) and∣∣h m Φ ′ (u m )ζ ∣ ∥ ∥Φ ′∥ ∥ ∞|h|ζ ∈ L 1 (Ω)by the assumption h ∈ L 1 χ.AcknowledgmentsThe author is thankful to Prof. G.I. Barenblatt for drawing the attention of Prof. H. Brezis to thisinteresting problem. He also thanks Prof. Brezis for stimulating discussions, and Prof. Vogelius for hisinterest and useful references.References G.I. Barenblatt, J.B. Bell and W.Y. Crutchfield, The thermal explosion revisited, Proc. Nat. Acad. Sci. USA 95 (1998),13384–13386. H. Brezis, Is there failure of the inverse function theorem?, in: Proceedings of the Workshop Held at the MorningsideCenter of Mathematics, Chinese Academy of Science, Beijing, June 1999. H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(1998), 223–262. H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for u t −∆u = g(u) revisited, Adv. Differential Equations1 (1996), 73–90. H. Brezis and J.L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10(1997), 443–469. G. Buttazzo, G. Dal Maso and U. Mosco, Asymptotic behaviour for Dirichlet problems in domains bounded by thinlayers, in: Partial Differential Equations and the Calculus of Variations, Vol. I, Birkhäuser Boston, Boston, MA, 1989,pp. 193–249. D. Cioranescu and F. Murat, A strange term coming from nowhere, in: Topics in the Mathematical Modelling of CompositeMaterials, Birkhäuser Boston, Boston, MA, 1997, pp. 45–93.

J. Dávila / A nonlinear elliptic equation with rapidly oscillating boundary conditions 307 G. Dal Maso and A. Garroni, New results on the asymptotic behavior of Dirichlet problems in perforated domains, Math.Models Methods Appl. Sci. 4 (1994), 373–407. G. Dal Maso and A. Garroni, The capacity method for asymptotic Dirichlet problems, Asymptotic Anal. 15 (1997), 299–324. G. Dal Maso and U. Mosco, Wiener’s criterion and Γ -convergence, Appl. Math. Optim. 15 (1987), 15–63. A. Damlamian, Le problème de la passoire de Neumann, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), 427–450. J. Dávila, A strong maximum principle for the Laplace equation with mixed boundary condition, J. Funct. Anal. 183(2001), 231–244. H. Fujita, On the nonlinear equations ∆u + e u = 0and∂v/∂t = ∆v + e v , Bull. Amer. Math. Soc. 75 (1969), 132–135. I.M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295–381. P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966),271–310. Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math. 23 (1997), 161–168. G. Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie Hölderiane, Ann. Mat. PuraAppl. (4) 51 (1960), 1–37.

More magazines by this user
Similar magazines