An elliptic problem with a layer

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Veröffentlicht in:	Нелинейные граничные задачи
Datum:	1999
1. Verfasser:	Biroli, M.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 1999
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spelling	Biroli, M. 2020-06-09T16:17:50Z 2020-06-09T16:17:50Z 1999 An elliptic problem with a layer / M. Biroli // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 13-22. — Бібліогр.: 6 назв. — англ. 0236-0497 https://nasplib.isofts.kiev.ua/handle/123456789/169266 en Інститут прикладної математики і механіки НАН України Нелинейные граничные задачи An elliptic problem with a layer Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	An elliptic problem with a layer
spellingShingle	An elliptic problem with a layer Biroli, M.
title_short	An elliptic problem with a layer
title_full	An elliptic problem with a layer
title_fullStr	An elliptic problem with a layer
title_full_unstemmed	An elliptic problem with a layer
title_sort	elliptic problem with a layer
author	Biroli, M.
author_facet	Biroli, M.
publishDate	1999
language	English
container_title	Нелинейные граничные задачи
publisher	Інститут прикладної математики і механіки НАН України
format	Article
issn	0236-0497
url	https://nasplib.isofts.kiev.ua/handle/123456789/169266
citation_txt	An elliptic problem with a layer / M. Biroli // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 13-22. — Бібліогр.: 6 назв. — англ.
work_keys_str_mv	AT birolim anellipticproblemwithalayer AT birolim ellipticproblemwithalayer
first_indexed	2025-11-25T22:34:36Z
last_indexed	2025-11-25T22:34:36Z
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fulltext	AN ELLIPTIC PROBLEM WITH A LAYER c© M. Biroli 1. Introduction. We are interested in the following problem: let Ω be an open set in RN we denote by Σ the set Ω ∩ {xN = 0} and we assume Σ 6= ∅. Moreover we denote x = (x′, xN ) = (x1, x2, ..., xN−1, xN ). We consider the problem ∫ DuDv m(dx) + ∫ D′uD′v σ(dx) = 0 (1.1) u ∈ H1 loc(Ω) with trace in H1 loc(Σ) ∀v ∈ H1 loc(Ω) with trace in H1 loc(Σ), with supp(v) ⊆ Ω where m denotes the Lebesgue measure on RN and σ denotes the Lebesgue measure on RN−1; moreover we denote by D the gradient in RN and by D′ the tangential gradient on RN−1 x′ . If u verify (1.1) we say that u is a solution of (1.1). If we replace in (1.1) the equality by the inequality ≤ (≥) and we consider only positive test functions v we say that u is a subsolution (supersolution) of (1.1) in Ω. The aim of this paper is to study the local regularity for a solution of (1.1). If we consider a ball that does not intersect Σ the problem of the regularity of u reduce to the problem of the regularity of an harmonic function; then in particular Harnack inequality for nonegative u and Hölder continuity for u hold. Problems arise in the case of sets having a non empty intersection with Σ (due to the different rescaling by the usual dilation of the two terms in (1.1)). We also observe that the bilinear form in (1.1) defines a strongly local regular Dirichlet form on L2(Ω,m + δΣ),[3], but the measure m + δΣ does not verify a doubling property then the regularity theory in [1][2] does not apply. To study the local regularity of u in B(x0, r) with, x0 ∈ RN−1 x′ we modify the def- inition of a ball defining B(x0, r) = {x : \|x′ − (x0)′\|4 + \|xN − (x0)N \|2 < r4} and we write S(x0, r) = B(x0, r) ∩ {xN = 0}. We define a cut-off function between B(x0, tr) and B(x0, sr), s, t ∈ [ 12 , 1) s < t, as η(x) = φ(d(x − x0)) where d(x) = (\|x′\|4 + x2 N ) 1 4 , φ(ρ) = 1 for ρ ≤ sr, φ(ρ) = 0 for ρ ≥ tr, 0 ≤ φ ≤ 1 and \|φ′\| ≤ C (t−s)r . Then \|D′η\| ≤ C (t− s)r on S(x0, r), \|Dη\| ≤ C (t− s)r2 on B(x0, r). With such a modification we obtain: Theorem 1.1. Let u be a nonegative solution of (1.1) in B(x0, 4r), x0 ∈ RN−1 x′ ; then supB(x0,r)u ≤ C infB(x0,r)u where C is a constant depending only on N. Theorem 1.1. Let u be a solution of (1.1) in Ω; then u is locally Hölder continuous in Ω. In Section 2 we prove suitable Poincaré and Sobolev type inequalities, that play a fundamental role in the proof, by a Moser type iteration method, of local L∞ estimates for solutions (or subsolutions) of (1.1), given in section 3. In section 4 we prove the results in Theorems 1.1 and 1.2. The result in Theorem 1.2 is an easy consequence of the one in Theorem 1.1; the proof of the result in Theorem 1.1 uses an iteration method introduced by Moser,[6], that allow use to consider estimates only on concentric balls B(x0, r); this last opportunity is usefull due to the different forms of the balls in the case x0 ∈ Σ or x0 /∈ Σ. 2. Poincaré and Sobolev type inequalities. It is well known that a fundamental role in the local regularity theory of harmonic functions relative to an uniformly elliptic operator is played by the usual Poncaré and Sobolev inequalities. The goal of this section is to prove suitably adapted Poincaré and Sobolev inequalities relative to the problem in consideration. Proposition 2.1. Let u be a function in H1(B(x0, r)), x0 ∈ RN−1 x′ , with a trace in H1(S(x0, r)) and ur be the average of u on S(x0, r) relative to the measure σ; then ∫ B(x0,r) \|u− ur\|2 m(dx) + ∫ S(x0,r) \|u− ur\|2 σ(dx) ≤ ≤ C[r4 ∫ B(x0,r) \|DNu\|2 m(dx) + r2 ∫ S(x0,r) \|D′u\|2 σ(dx)] where ∫ B(x0,r) m(dx) ( ∫ S(x0,r) σ(dx)) denotes the average on the set B(x0, r) (S(x0, r)) relative to the measure m (σ). The result in Proposition 2.1 isa consequence of the following Sobolev type inequality: Proposition 2.2. Let u be a function in H1(B(x0, r)), x0 ∈ RN−1 x′ , with a trace in H1(S(x0, r)). (a) Let N > 3; there exists q > 2 such that [ ∫ B(x0,r) \|u− ur\|q m(dx) + ∫ S(x0,r) \|u− ur\|q σ(dx)] 2 q ≤ ≤ C[r4 ∫ B(x0,r) \|DNu\|2 m(dx) + r2 ∫ S(x0,r) \|D′u\|2 σ(dx)] (b) Let N = 3; for every q > 2 we have [ ∫ B(x0,r) \|u− ur\|q m(dx) + ∫ S(x0,r) \|u− ur\|q σ(dx)] 2 q ≤ ≤ C[r4 ∫ B(x0,r) \|DNu\|2 m(dx) + r2 ∫ S(x0,r) \|D′u\|2 σ(dx)] (c) Let N = 2; then oscB(x0,r)u ≤ ≤ C[r4 ∫ B(x0,r) \|DNu\|2 m(dx) + r2 ∫ S(x0,r) \|D′u\|2 σ(dx)] 1 2 Proof. We prove the result for the case (a); the proof in the cases (b) and (c) is analo- gous. It is enough to prove the result in the case r = 1 and we write B(x0, r) = B, S(x0, r) = S. Let s = 2N−2 N−3 , we have ∫ S \|u− u1\|s σ(dx)] 2 s ≤ ≤ C ∫ S \|D′u\|2 σ(dx), where we denote by C possibly different constants depending only on N . Let q = 2+s 2 = 2N−4 N−3 , by easy computations we obtain sup((x0)N−1,(x0)N+1) ∫ B∩{xN=t} \|u− u1\|q σ(dx) ≤ ≤ C[( ∫ B \|DNu\|2 m(dx)) q 2 + ( ∫ S \|u− u1\|s σ(dx)) q s ] ≤ ≤ C[( ∫ B \|DNu\|2 m(dx)) q 2 + ( ∫ S \|D′u\|2 σ(dx)) q 2 ] and the result follows. 3. The local L∞ estimate for subsolutions. We prove at first an L∞ estimate for nonegative subsolution of (1.1) and finally we prove the general L∞ estimate for nonegative solutions of (1.1) Proposition 3.1. Let u be a function in H1(B(x0.r)), x0 ∈ RN−1 x′ , with a trace in H1(S(x0, r)). Assume that u is a nonnegative subsolution in a neighbourhood of B(x0, r); then there exists constants d and C such that for α ∈ [ 12 , 1) and p ≥ 2 we have (supB(x0,αr)u)p ≤ C (1− α)d [ ∫ B(x0,r) upm(dx) + ∫ S(x0,r) upσ(dx)] 1 p Proof. Let β ≥ 1 and 0 < M < +∞; we define HM (t) = tβ for t ∈ [0,M ] HM (t) = Mβ + βMβ−1(t−M) for t > M The function HM (t) is Lipschitz-continuous for every fixed M . We assume that u is Lipschitz continuous (if it is not the case we use an approximation of u in H1(B(x0, r)) and in H1(S(x0, r)) by a sequence {uk} of nonegative Lipschitz- continuous functions). For a fixed M we define φ(x) = η(x)2 ∫ u(x) 0 H ′ M (t)2dt where η is a Lipschitz continuous function with support in B(x0, r) to be choosen. We observe that φ are nonegative Lipschitz continuous functions defined in B(x0, r) and Diφ = η2H ′ M (u)2Diu + 2ηDiη ∫ u(x) 0 H ′ M (t)2dt (3.1) for i = 1, 2, ..., N . Since u is a subsolution we have ∫ Du η2H ′ M (u)2Du m(dx)+ (3.2) + ∫ Du 2ηDη( ∫ uk 0 H ′ M (t)2dt) m(dx) + ∫ Du η2H ′ M (u)2Du σ(dx)+ + ∫ Du 2ηDη( ∫ u 0 H ′ M (t)2dt) σ(dx) ≤ 0. We observe that \|Diu 2ηDiη( ∫ u 0 H ′ M (t)2dt)\| ≤ ≤ 1 2 Diu η2H ′ M (u)2Diu+ +2\|Diη\|2( 1 H ′ M (u) ∫ u 0 H ′ M (t)2dt)2 ≤ ≤ 1 2 Diu η2H ′ M (u)2Diu + +2\|Diη\|2(uH ′ M (u))2 From (3.2) it follows 1 2 ∫ \|D(HM (u))\|2η2 m(dx)+ (3.3) + 1 2 ∫ \|D(HM (u))\|2η2 σ(dx) ≤ ≤ 2 ∫ \|Dη\|2(uH ′ M (u))2 m(dx) + 2 ∫ \|Dη\|2(uH ′ M (u))2 σ(dx). We choose now η as the cut-off function between the B(x0, sr) and B(x0, tr). From (3.3) we obtain ∫ B(x0,sr) \|D(HM (u))\|2 m(dx)+ (3.4) + ∫ S(x0,sr) \|D(HM (u))\|2 σ(dx) ≤ ≤ c2 (t− s)2r4 ∫ B(x0,tr) (uH ′ M (u))2 m(dx)+ + c2 (t− s)2r2 ∫ S(x0,tr) (uH ′ M (u))2 σ(dx). Using the Sobolev inequality we obtain [ ∫ B(x0,sr) \|HM (u)− (HM (u))sr\|q m(dx)+ + ∫ S(x0,sr) \|HM (u)− (HM (u))sr\|q σ(dx)] 1 q ≤ c3 s t− s [ ∫ B(x0,tr) (uH ′ M (u))2 m(dx) + ∫ S(x0,tr) (uH ′ M (u))2 σ(dx)] 1 2 . We use the inequality H(t) ≤ tH ′(t) and we obtain [ ∫ B(x0,sr) \|HM (u)\|q m(dx) + ∫ S(x0,sr) \|HM (u)\|q σ(dx)] 1 q ≤ c5( s t− s + 1)[ ∫ B(x0,tr) (uH ′ M (u))2 m(dx)+ + ∫ S(x0,tr) (uH ′ M (u))2 σ(dx)] 1 2 . We observe that ( s t−s + 1) ≤ 2 s t−s . We take into account the definition of HM and we let M → +∞; then [ ∫ B(x0,sr) uβq m(dx) + ∫ S(x0,sr) uβq σ(dx)] 1 q ≤ c6β s t− s [ ∫ B(x0,tr) u2β m(dx) + ∫ S(x0,tr) u2β σ(dx)] 1 2 . We write 2β = ν, q = 2τ (τ > 1) and we obtain [ ∫ B(x0,sr) uτν m(dx) + ∫ S(x0,sr) uτν σ(dx)] 1 τν ≤ (3.5) (c6ν s t− s ) 2 ν [ ∫ B(x0,tr) uν m(dx) + ∫ S(x0,tr) uν σ(dx)] 1 ν . From (3.5) an iteration method of Moser’s type (see for example [3]) give the result. Proposition 3.2. Let u be a local nonegative solution of our problem in B(x0, 2r), x0 ∈ RN−1 x′ . Then there exists constants d, τ > 1 and C such that for α ∈ [ 12 , 1) and every real p we have (supB(x0,αr)u)p ≤ ≤ C (1− α)d (1 + \|p\|) 2τ τ−1 [ ∫ B(x0,r) upm(dx) + ∫ S(x0,r) upσ(dx)] 1 p Proof. It is enough to prove the result in the case −∞ < p < 2 and u ≥ ε > 0. By Proposition 3.1 u is bounded in B(x0, r); we define φ = η2uβ ,with β ≤ 1 and we can prove that φ is in H1(B(x0, r)) and its trace is in H1(S(x0, r)). We recall that Diφ = η2βuβ−1Diu + 2ηDiη uβ Di(u β+1 2 ) = β + 1 2 u β−1 2 Diu. Then for β 6= −1 we obtain \| β β + 1 \|2 ∫ B(x0,r) \|D(u β+1 2 )\|2η2 m(dx)+ (3.6) +\| β β + 1 \| ∫ S(x0,r) \|D′(u β+1 2 )\|2η2 σ(dx)) ≤ ≤ ∫ B(x0,r) \|D(u β+1 2 )Dη\|u β+1 2 η m(dx)+ + ∫ S(x0,r) \|D′(u β+1 2 )Dη\|u β+1 2 η σ(dx). From (3.6) we easily obtain for β 6= 0,−1 ∫ B(x0,r) \|D(u β+1 2 )\|2η2 m(dx) + ∫ S(x0,r) \|D′(u β+1 2 )\|2η2 σ(dx) ≤ ≤ ( β + 1 β )2\| ∫ B(x0,r) \|Dη\|2uβ+1 m(dx)+ +( β + 1 β )2 ∫ S(x0,r) \|D′η\|2uβ+1 σ(dx). Then, taking again η as the cut-off function between B(x0, sr) and B(x0, tr), 1 2 ≤ s < t < 1, we have ∫ B(x0,sr) \|D(u β+1 2 )\|2 m(dx) + ∫ S(x0,sr) \|D′(u β+1 2 )\|2 σ(dx) ≤ ≤ ( β + 1 β )2 1 (t− s)2r4 ∫ B(x0,tr) uβ+1 m(dx)+ +( β + 1 β )2 1 (t− s)2r2 ∫ S(x0,tr) uβ+1 σ(dx). We use now the Sobolev inequality in Proposition 2.2; then by the same methods as in Proposition 3.1 we have [ ∫ B(x0,sr) u β+1 2 q m(dx) + ∫ S(x0,sr) u β+1 2 q σ(dx)] 1 q ≤ (3.7) ≤ c\|β + 1 β \|( s t− s + 1)[( ∫ B(x0,tr) uβ+1 m(dx)+ + ∫ S(x0,tr) uβ+1 σ(dx)] 1 2 . Setting β + 1 = ν and q = 2τ we have for any −∞ < ν ≤ 2, ν 6= 0,−1 [ ∫ B(x0,sr) uτν m(dx) + ∫ S(x0,sr) uτν σ(dx)] 1 τ\|ν\| ≤ (3.8) ≤ c 2 \|ν\| (\| ν ν − 1 \| s t− s + 1) 2 \|ν\| [( ∫ B(x0,tr) uν m(dx)+ + ∫ S(x0,tr) uν σ(dx)] 1 \|ν\| . From (3.8) the results follows by a Moser’s type iteration argument (see for example [3]). Proposition 3.3. Let the assumptions of Proposition 3.2 hold and assume that u ≥ ε > 0. For α ∈ [ 12 , 1) define k by logk = ∫ S(x0,αr) logu σ(dx), x0 ∈ Σ; then for λ > 0 we have m{x ∈ B(x0, αr); \|log( u(x) k )\| > λ} ≥ C 1− α m(B(x0, αr)) σ{x ∈ S(x0, αr); \|log( u(x) k )\| > λ} ≥ C 1− α σ(S(x0, αr)) where C is a constant that does not gepend on ε. Proof. By easy computations we obtain ∫ \|D(logu)\|2η2 m(dx) + ∫ \|D′(logu)\|2η2 σ(dx) ≤ ≤ 4 ∫ \|Dη\|2 m(dx) + ∫ \|D′η\|2 σ(dx) ≤ ≤ c (1− α)2 ( m(B(x0, r)) r4 + σ(S(x0, r)) r2 ). where η is the cut-off function between B(x0, αr) and B(x0, r). By Proposition 2.1 we obtain ∫ B(x0,αr) \|logu− logk\|2 m(dx) ≤ c (1− α)2 m(B(x0, r)) r4 (3.9) ∫ S(x0,αr) \|logu− logk\|2 σ(dx) ≤ c (1− α)2 σ(S(x0, r)) r2 ). (3.10) From (3.10) and (3.11) the result easily follows. 4. Proof of Theorems 1.1 and 1.2. We are now in position to prove Theorem 1.1. Lemma 4.1. Let m, µ, C, θ ∈ [ 12 , 1) be positive constants and let w > 0 be a function in H1(B(x0, r)), x0 ∈ RN−1 x′ , such that supB(x0,sr)w p ≤ (4.1) C (t− s)d 1 m(B(x0, r)) ∫ B(x0,tr) wp m(dx)+ + C (t− s)d 1 σ(S(x0, r)) ∫ S(x0,tr) wp σ(dx) for all 1 2 ≤ θ ≤ s < t ≤ 1, 0 < p < µ−1. Moreover, let m(x ∈ B(x0, r); logw ≥ λ) ≤ Cµ λ m(B(x0, r)) (4.2) σ(x ∈ S(x0, r); logw ≥ λ) ≤ Cµ λ σ(B(x0, r)) (4.3) for all λ > 0. Then there exists a constant γ = γ(θ, d, C) such that supB(x0,θr)u ≤ γµ Proof. We assume, without loss of generality r = 1. Replacing w by wµ and λ by λµ we reduce us to the case µ = 1. Define φ(s) = supB(x0,s)logw, θ ≤ s < 1; we observe that φ(s) is a nondecreasing function. We now prove that the following inequality holds: φ(s) ≤ 3 4 φ(t) + γ1 (t− s)2d (4.4) where θ ≤ s < t ≤ 1 and γ1 is a constant depending on θ, d, C. We decompose B(x0, t) and S(x0, t) into the sets where logw > 1 2φ(t) and where logw ≤ 1 2φ(t); then taking into account (4.2) and (4.3) we obtain ∫ B(x0,t) wp m(dx) ≤ (epφ(t) 2C φ(t) + ep φ(t) 2 )m(B(x0, 1)) ∫ S(x0,t) wp σ(dx) ≤ (epφ(t) 2C φ(t) + ep φ(t) 2 )σ(S(x0, 1)). Summing up the two inequalities we have 1 m(B(x0, 1)) ∫ B(x0,t) up m(dx) + 1 σ(S(x0, 1)) ∫ S(x0,t) up σ(dx) ≤ ≤ 2(epφ(t) 2C φ(t) + ep φ(t) 2 ). We choose now p such that the two terms in the right-hand side are equal: p = 2 φ(t) log( φ(t) 2C ) provided the term in the right-hand side is less than µ−1 = 1; this last inequality requires φ(t) > c1 (4.5) where c1 depends only on C. In that case we have 1 m(B(x0, 1)) ∫ B(x0,t) up m(dx) + 1 σ(S(x0, 1)) ∫ S(x0,t) up σ(dx) ≤ ≤ 4ep φ(t) 2 , hence by (4.1) we obtain φ(s) ≤ 1 p log( 4C (t− s)d ep φ(t) 2 ) = = 1 p log( 4C (t− s)d ) + 1 2 φ(t) Then, taking into account the fixed value of p, the above inequality becomes φ(s) ≤ 1 2 φ(t){ log( 4C (t−s)d ) log(φ(t) 2C ) ) + 1}. If φ(t) ≥ 32C3 (t− s)2d (4.6) we obtain φ(s) ≤ 3 4 φ(t) then (4.4) holds. If (4.5) or (4.6) does not hold; then φ(s) ≤ φ(t) ≤ γ1 (t− s)2d , θ ≤ s < t ≤ 1, where γ1 is a constant depending on C, c1, d, θ; so (4.4) holds again. We have so proved the inequality (4.4); the result now follows by iteration as in Lemma 3 in [6]. We are now in position to prove the result of Theorem 1.1. We assume, without loss of generality, u ≥ ε > 0. We use the result in Proposition 4.1 for u k and k u ; Where logk = ∫ S(x0,r) logu σ(dx). The assumptions of proposition 4.1 hold in B(x0, 2r) by Proposition 3.2 and 3.3; then supB(x0,r) u k ≤ C, supB(x0,r) k u ≤ C Then the result follows. The result in Theorem 1.2 can be proved from Theorem 1.1 by standard methods (see [5]). References 1. M. Biroli, U. Mosco, Formes de Dirichlet etestimations structurelles dans les milieux discontinus, C.R.A.S. Paris 313 (1991), 593–598. 2. M. Biroli, U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. di Mat. Pura e Appl. CLXIX (IV) (1995), 125–181. 3. S. Chanillo, R.L. Wheeden, Harnack’s inequality and mean-value inequalities for solutions of de- generate elliptic equations, Comm. in P.D.E. 11 (10) (1986), 1111–1134. 4. M. Fukushima, Dirichlet forms and Markov processes, North-Holland, Amsterdam, 1980. 5. D. Gilbarg, N.S. Trudinger, Second order elliptic equations, Grundlehren der Math. Wiss. 224, Springer Verlag, Berlin-Heidelberg-New York, 1977. 6. J. Moser, On a pointwise estimate for parabolic differential equations, Com. in Pure and Appl. Math. XXIV (1971), 727–740. Department of Mathematics Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano (Italy) E-mail: marbir@mate.polimi.it

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