BOUNDARY REGULARITY FOR THE POISSON EQUATION IN REIFENBERG-FLAT DOMAINS. Antoine Lemenant. Yannick Sire

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1 BOUNDARY REGULARITY FOR THE POISSON EQUATION IN REIFENBERG-FLAT DOMAINS Antoine Lemenant LJLL, Univesité Pais-Dideot, CNRS Pais, Fance Yannick Sie LATP, Univesité Aix-Maseille, CNRS Maseille, Fance Abstact This pape is devoted to the investigation of the bounday egulaity fo the Poisson equation { u = f in u = on whee f belongs to some L p ( and is a Reifenbeg-flat domain of R n Moe pecisely, we pove that given an exponent α (,, thee exists an ε > such that the solution u to the pevious system is locally Hölde continuous povided that is (ε, -Reifenbeg-flat The poof is based on Alt-Caffaelli- Fiedman s monotonicity fomula and Moey-Campanato theoem The goal of the pesent pape is to pove a bounday egulaity esult fo the Poisson equation with homogeneous Diichlet bounday conditions in non smooth domains We conside the case of Reifenbeg-flat domains as given by the following definition Definition Let ε, be two eal numbes satisfying < ε < / and > An (ε, -Reifenbeg-flat domain R N is an open, bounded, and connected set satisfying the two following conditions : (i fo evey x and fo any, thee exists a hypeplane P (x, containing x which satisfies d H( B(x,, P (x, B(x, ε (ii fo evey x, one of the connected component of B(x, {x; d(x, P (x, ε } is contained in and the othe one is contained in c 99 Mathematics Subject Classification Pimay: ; Seconday: Key wods and phases Elliptic poblem in nonsmooth domain, Reifenbeg-flat domains, Regulaity

2 ANTOINE LEMENANT AND YANNICK SIRE We conside the following poblem in the (ε, -Reifenbeg-flat domain R N fo some f L q (, { u = f in (P u = on Reifenbeg-flat domains ae less smooth than Lipschitz domains and it is well known that we cannot expect moe egulaity than Hölde fo bounday egulaity of the Poisson equation in Lipschitz domains (see [9], [8, Remak 7], o [] Histoically, Reifenbeg-flat domains came into consideation because of thei elationship with the egulaity of the Poisson kenel and the hamonic measue, as shown in a seies of famous and deep papes by Kenig and Too (see fo eg [, 3, 4, ] In paticula, they ae Non Tangentially Accessible (in shot NTA domains as descibed in [] Notice that the Poisson kenel is defined as elated to the solution of the equation u = in with u = f on In this pape we conside the equation (P which is of diffeent natue Howeve, ou egulaity esult is again based on the monotonicity fomula of Alt, Caffaelli and Fiedman [], which is known to be one of the key estimate in the study hamonic measue as well Moe ecently, egulaity of elliptic PDEs in Reifenbeg-flat domains has been studied by Byun and Wang in [6, 5,, 3, 4] (see also the efeences theein One of thei main esult egading to equation of the type of (P is the existence of a global W,p ( bound on the solution This fact will be used in Coollay below The case of domains of R n has been investigated by Caffaelli and Peal [7] See also [] fo the case of Lipschitz domains Some othe type of elliptic poblems in Reifenbeg-flat domains can be found in [6, 7, 5, 9, 8] The pesent pape is the fist step towads a geneal bounday egulaity theoy fo elliptic PDEs in divegence fom on Reifenbeg-flat domains, that might be pusued in some futue wok Ou main esult is the following Theoem Let p, q, p be some exponents satisfying p + q = p and p > N Let α > be any given exponent such that ( α < p N p Then one can find an ε = ε(n, α such that the following holds Let R N be an (ε, -Reifenbeg-flat domain fo some >, and let u be a solution fo the poblem (P in with u L p ( and f L q ( Then u C,α (B(x, / x Moeove u C,α (B(x, / C(N,, α, p, u p, f q Obseve that in the statement of Theoem, some a pioi L p integability on u is needed to get some Hölde egulaity In what follows we shall see at least two situations whee we know that u L p fo some p >, and consequently state two Coollaies whee the integability hypothesis is given on f only, without any a pioi equieement on u

3 REGULARITY IN REIFENBERG FLAT DOMAINS 3 Fist, notice that when f L q ( fo q +, then the application v, vfdx is a bounded Linea fom on W (, endowed with the scala poduct u v dx Theefoe, using Riesz epesentation Theoem we deduce the existence of a unique weak solution u W, ( fo the poblem (P Moeove, the Sobolev inequality says that u L (, with = N Some simple computations shows that in this situation, u and f veify the statement of Theoem povided that N 5, which leads to the following coollay Coollay Assume that N 5 and let q I N be given whee { [, + if N = I N =, + fo 3 N 5 Then fo any α > veifying ( N 6 N α < N ( N N +, q we can find an ε = ε(n, α such that the following holds Let R N be an (ε, - Reifenbeg-flat domain fo some >, let f L q ( and let u W, ( be the unique solution fo the poblem (P in Then u C,α (B(x, / x Moeove u C,α (B(x, / C(N,, α, q, u, f q Poof Since u W, (, the Sobolev embedding says that u L p (, with p = = N And by assumption f Lq fo some q I N (notice that q, which guaantees existence and uniqueness of the weak solution We now ty to apply Theoem with those p and q Let p be defined by p + q = p Then a simple computation yields that p > N, povided that q > N 6 N This fixes the ange of dimension N 5 and notice that in this case N 6 N except fo N =, which justifies the definition of I N We then conclude by applying Theoem In the poof of Coollay we butally used the Sobolev embedding on W, ( to obtain an L p integability on u But unde some natual hypothesis we can get moe using a Theoem by Byun and Wang [] Pecisely, if f = divf fo some F L, then f lies in the dual space of H ( which guaantees the existence and uniqueness of a weak solution u fo (P, again by the Reisz epesentation Theoem The theoem of Byun and Wang [, Theoem ] implies moeove that if F L ( then u L ( as well But then the Sobolev inequality says that u L which allows us to apply Theoem fo a lage ange of dimensions and exponents Of couse this analysis is inteesting only fo N because if > N we

4 4 ANTOINE LEMENANT AND YANNICK SIRE diectly get some Hölde estimates by the classical Sobolev embedding This leads to ou second coollay Coollay Let N 3 Then fo any α > satisfying < N and q > N 3 N α < N ( +, with := N q N, be given, so that moeove we can find an ε = ε(n, α,, such that the following holds Let R N be an (ε, -Reifenbeg-flat domain fo some >, let f L q ( and assume that f = divf fo some F L ( Let u W, ( be the unique weak solution fo the poblem (P in Then u C,α (B(x, / x Moeove u C,α (B(x, / C(N,, α,, q,, f q, F Poof Fist we apply [, Theoem ] which povides the existence of a theshold ε = ε(n,, such that fo any solution u W, ( of (P with f = divf and F L (, we have that u L (, povided that is (ε, -Reifenbegflat But then the Sobolev inequality implies that u L with = N if < N, N and = + othewise In ode to apply Theoem we define p such that + q = p, and we only need to check that p < N This implies the following condition on and q : > N 3 and q > N 3 N, as equied in the statement of the Coollay We finally conclude by applying Theoem Ou appoach to pove Theoem follows the one that was aleady used in [8] to contol the enegy of eigenfunctions nea the bounday of Reifenbeg-flat domains, and that we apply hee to othe PDE than the eigenvalue poblem The main ingedient in the poof is a vaiant of Alt-Caffaelli-Fiedman s monotonicity fomula [, Lemma 5], to contol the behavio of the Diichlet enegy in balls centeed at the bounday, as well as in the inteio of Then we conclude by Moey-Campanato Theoem The monotonicity Lemma We begin with a technical Lemma which basically contains the justification of an integation by pats The poof is exactly the same as the fist step of [8, Lemma 5] but we decided to povide hee the full details fo the convenience of the eade

5 REGULARITY IN REIFENBERG FLAT DOMAINS 5 Lemma Let u be a solution fo the poblem (P Then fo evey x and ae > we have x N dx N u u B(x, B(x, ν ds (N ( + N u ds + uf x N dx B(x, B(x, Poof Although (3 can be fomally obtained though an integation by pats, the igoous poof is a bit technical In the sequel we use the notation := B(x,, and S + := B(x, We find it convenient to define, fo a given ε >, the egulaized nom x ε := x + x + + x N + ε, so that x ε is a C function A diect computation shows that ( x ε N = ( NN ε x N+, ε in othe wods x N ε is supehamonic, and hopefully enough this goes in the ight diection egading to the next inequalities We use one moe egulaization thus we let u n Cc ( be a sequence of functions conveging in W, (R N to u We now poceed as in the poof of Alt, Caffaelli and Fiedman monotonicity fomula []: by using the equality (3 we deduce that (4 u n x N ε = (u n = u n + u n u n (u n x N ε (u n u n x N ε Since ( x N ε, the Gauss-Geen Fomula yields (5 (u n x N ε dx = u n ( x N ε dx + I n,ε ( I n,ε (, whee I n,ε ( = ( + ε N In othe wods, (4 eads (6 u n x N ε dx I n,ε ( u n u n ds + (N ν ( + ε N (u n u n x N ε dx u nds We now want to pass to the limit, fist as n +, and then as ε + To tackle some technical poblems, we fist integate ove [, + δ] and divide by δ, thus obtaining (7 whee δ +δ ( + ρ u n x N ε dx A n = δ +δ dρ A n R n, I n,ε (ρdρ

6 6 ANTOINE LEMENANT AND YANNICK SIRE and R n = δ +δ ( + ρ (u n u n x N ε dx Fist, we investigate the limit of A n as n + : by applying the coaea fomula, we ewite A n as ( A n = ( x +ε N x u n u n x dx+( u δ ndx +δ \+ +δ \+ ( x + ε N Since u n conveges to u in W, (R N when n +, then by using again the coaea fomula we get that whee A n δ I ε (ρ = (ρ + ε N + ρ +δ I ε (ρdρ n +, dρ u u ρ ds + (N ν (ρ + ε N + ρ u ds Next, we investigate the limit of R n as n + By using Fubini s Theoem, we can ewite R n as R n = (u n u n G(xdx, whee G(x = x N +δ ε δ + ρ (xdρ Since if x +δ + + δ x δ + ρ (xdρ = if x +δ δ \ +, if x / +δ then G is Lipschitz continuous and hence by ecalling u n Cc ( we get ( ( u n u n u u Gdx ( u n u u n Gdx + u n u ugdx ( ( ( = u n u u n G + u n G dx + u ( u n ug + (u n u G dx ( u n u L ( u n L ( G L ( + u n L ( G L ( + u L ( ( u n u L ( G L ( + u n u L ( G L ( and hence the expession at the fist line conveges to as n + By combining the pevious obsevations and by ecalling that u satisfies the equation in the poblem (P we infe that by passing to the limit n in (7 we get δ +δ ( + ρ x N ε dx d δ +δ I ε (ρdρ+ δ +δ ( + ρ uf x N ε dx dρ

7 REGULARITY IN REIFENBERG FLAT DOMAINS 7 Finally, dividing by, by passing to the limit δ +, and then ε + obtain (3 we Next, we will need the following Lemma of Gonwall type Lemma Let γ >, >, ψ : (, R be a continuous function and ϕ : (, R be an absolutely continuous function that satisfies the following inequality fo ae (,, (8 ϕ( γϕ ( + ψ( Then ϕ( + γ γ is a nondeceasing function on (, ψ(s ds s + γ Poof We can assume that ψ(s ds < +, s + γ othewise the Lemma is tivial Unde ou hypothesis, the function F ( := ϕ( + ψ(s ds γ γ s + γ is diffeentiable ae and absolutely continuous A computation gives thus (8 yields F ( = ϕ ( γ ϕ( γ γ which implies that F is nondecesing γ = ϕ ( ϕ( γ γ + ψ( γ + ψ( γ + γ + γ γf ( = γϕ ( ϕ( + ψ(, γ We now pove the monotonicity Lemma, which is inspied by Alt, Caffaelli and Fiedman [, Lemma 5] The following statement and its poof, is an easy vaiant of [8, Lemma 5], whee the same estimate is pefomed on Diichlet eigenfunctions of the Laplace opeato We decided to wite the full details in ode to enlighten the ole of the second membe f in the inequalities Lemma 3 Let R N be a bounded domain and let u be a solution fo the poblem (P Given x and a adius >, we denote by := B(x,, by S + := B(x, and by σ( the fist Diichlet eigenvalue of the Laplace opeato on the spheical domain S + If thee ae constants > and σ ], N [ such that (9 inf ( σ( σ, <<

8 8 ANTOINE LEMENANT AND YANNICK SIRE then the function ( ( β ψ(s dx + β ds x x s+β is non deceasing on ], [, whee β ], [ is given by and β = (N + 4σ (N ψ(s := + s uf x x N dx We also have the bound ( x x dx C(N,, β u L ( + ψ( / Remak Of couse the Lemma is inteesting only when ( ψ(s ds < + s+β This will be satisfied if u and f ae in some L p spaces with suitable exponents, as will be shown in Lemma Remak Notice that it is not known in geneal whethe u L p ( fo some p > and theefoe it is not obvious to find a bound fo the left hand side of ( Poof We assume without lose of geneality that x = and to simplify notation we denote by B the ball B(x, By Lemma we know that fo ae >, x N dx N u u (N ds + N u ds S + ν S + (3 + uf x N dx (4 Let us define ψ( := uf x N dx and assume that ( holds (othewise thee is nothing to pove Next, we point out that the definition of σ implies that u ds (5 σ τ u ds ], [, S + S + whee τ denotes the tangential gadient on the sphee Also, let α > be a paamete that will be fixed late, then by combining Cauchy-Schwaz inequality,

9 REGULARITY IN REIFENBERG FLAT DOMAINS 9 (5 and the inequality ab α a + α b, we get ( u u ( S + ν ds u ds u S + S + ν ds ( ( σ τ u ds u (6 Hence, (7 N S + ( α σ u u ds + (N N ν S + S + S + S + τ u ds + α u ds N σ ν ds S + u ν ds [ α S + + (N N σ [( α 3 N σ + N σ τ u ds + S + α σ Next, we choose α > in such a way that namely α + N σ σ = α σ, α = [ (N + 4σ (N ] σ Hence, by combining (3, (5 and (7 we finally get (8 x N dx 3 N γ(n, σ ds + ψ(, whee Let us set and obseve that hence (8 implies that γ(n, σ = S + S + [ ] (N + 4σ (N ϕ( := ϕ ( = N S + x N dx ae ], [, τ u ds + α S + S + u ] ν ds τ u ds u ] ν ds (9 ϕ( γϕ ( + ψ(, with γ = γ(n, σ But now Lemma exactly says that the function ( β dx + β x x ψ(s ds s+β

10 ANTOINE LEMENANT AND YANNICK SIRE is non deceasing on (,, whee β (, is given by β = γ = (N + 4σ (N, which poves the monotonicity esult To finish the poof of the Lemma it emains to establish ( Fo this pupose, we stat by finding a adius ( /, such that ds is less than S + aveage, which means ds dx u L S + ( By combining (8 with the fact that / we infe that x N dx C(N,, β u L ( + ψ( ( C(N,, β u L ( + ψ( It follows that / dx x u + dx x C(N,, β u L ( + ψ(, and ( is poved An elementay computation In ode to apply Lemma 3, the fist thing to check is that ( holds The pupose of the following Lemma is to pove that it is the case when u and f ae in suitable L p spaces Lemma Let R N be an abitay domain and let p > N Then fo any g L p (, denoting ψ( := g x N dx, we have ( As a consequence fo any β > satisfying (3 B(, ψ( C(N, p g p p N p ψ(s ds < + s+β β < p N p Poof Fist, we obseve that x N L m fo any m that satisfies (N m < N m < N N

11 REGULARITY IN REIFENBERG FLAT DOMAINS Moeove a computation gives that unde this condition, (4 x N m L m (B(, = x ( Nm = C(N, m N m( B(, Let us define m as being the conjugate exponent of p, namely m := p Then the hypothesis p > N N implies that m <, and by use of Hölde inequality and by (4 we can estimate ψ( g p x N L m (B(, C(N, m g p N m( m, o equivalently in tems of p, (5 ψ( C(N, p g p p N p The conclusion of the Lemma follows diectly fom (5 3 Inteio estimate We come now to ou fist application of Lemma 3, which is a decay estimate fo the enegy at inteio points, fo abitay domains Poposition Let p, q, p > be some exponents satisfying p + q = p and p > N Let β > be any given exponent such that β < p N (6 p Let R N be any domain, x, and let u be a solution fo the poblem (P in with u L p ( and f L q ( Then (7 dx C +β u p f q (, dist(x, /, B(x, with C = C(N, β, p, dist(x, Poof We assume that x = and we define := dist(x, in such a way that B(x, is totally contained inside fo and unde the notation of Lemma 3, = B(, and by S + = B(, fo any Unde ou hypothesis and in vitue of Lemma that we apply with g = uf, we know that ( holds fo any β > that satisfies (8 β < p N p Notice that p N p >, because p > N/ In the sequel we chose any exponent β > satisfying (8, so that ( holds and moeove Lemma says that (9 ψ(s s +β ds C(N, p u p f q C(N, p, β u p f q s p N p β ds

12 ANTOINE LEMENANT AND YANNICK SIRE We ae now eady to pove (7, β still being a fixed exponent satisfying (8 We ecall that the fist eigenvalue of the spheical Diichlet Laplacian on the unit sphee is equal to N, thus hypothesis (9 in the pesent context eads (3 inf ( σ( = N, << so that (9 holds fo any σ < N Let us choose σ exactly equal to the one that satisfies β = (N + 4σ (N one easily veifies that σ < N because of (8 As a consequence, we ae in position to apply Lemma 3 which ensues that, if u is a solution fo the poblem (P, then the function in ( is non deceasing In paticula, by monotonicity we know that fo evey, +β (3 dx ( ( β ( / β x x / dx + β dx x x and we conclude that fo evey /, dx K +β, with ( K = ( / β / ψ(s s +β ds / + β / ψ(s dx + β ds x x s+β ψ(s ds, s+β Let us now povide an estimate on K To estimate the fist tem in K we use ( to wite (3 x x dx C(N,, β u L ( + ψ( / Then we use ( to estimate ψ( C(N, p, u p f q, and fom the equation satisfied by u we get u L ( = ufdx u p f q so that in total we have x x dx C(N,, β, p u p f q / Finally, the last estimate togethe with (9 yields K C(N,, β, p u p f q, and this ends the poof of the Poposition

13 REGULARITY IN REIFENBERG FLAT DOMAINS 3 4 Bounday estimate We now use Lemma 3 again to povide an estimate on the enegy at bounday points, this time fo Reifenbeg-flat domains Poposition Let p, q, p > be some exponents satisfying p + q = p and p > N Let β > be any given exponent such that (33 β < p N p Then one can find an ε = ε(n, β such that the following holds Let R N be any (ε, -Reifenbeg-flat domain fo some >, let x and let u be a solution fo the poblem (P in with u L p ( and f L q ( Then (34 dx C +β u p f q (, /, B(x, with C = C(N,, β, p Poof As befoe we assume that x = and we denote by := B(, and by S + := B(, To obtain the decay estimate on + u dx we will follow the poof of Poposition : the main diffeence is that fo bounday points, (3 does not hold This is whee Reifenbeg-flatness will play a ole Let β >, be an exponent satisfying (33, so that invoquing Lemma we have (35 ψ(s s +β ds C(N, p, β u p f q < + Next, we ecall that the fist eigenvalue of the spheical Diichlet Laplacian on a half sphee is equal to N (as fo the total sphee Fo t (,, let S t be the spheical cap S t := B(, {x N > t} so that t = coesponds to a half sphee Let λ (S t be the fist Diichlet eigenvalue in S t In paticula, t λ (S t is continuous and monotone in t Theefoe, since and λ (S t as t, thee is t (β < such that β = (N + 4λ (t (N By applying the definition of Reifeneg flat domain, we infe that, if ε < t (η/, then B(x, is contained in a spheical cap homothetic to S t fo evey Since the eigenvalues scale of by facto when the domain expands of a facto /, by the monotonicity popety of the eigenvalues with espect to domains inclusion, we have (36 inf < λ ( B(x, λ (S t = β ( β + N As a consequence, we ae in position to apply the monotonicity Lemma (Lemma 3 which ensues that, if u is a solution fo the poblem (P and x, then the function in ( is non deceasing We then conclude as in the poof of Poposition

14 4 ANTOINE LEMENANT AND YANNICK SIRE, ie by monotonicity we know that fo evey <, ( +β dx β dx + β + x x ( (37 ( / β dx x x hence fo evey /, with, ( K = ( / β / / dx K +β, ψ(s s +β ds / + β / ψ(s dx + β ds x x s+β ψ(s ds, s+β Then we estimate K exactly as in the end of the poof of Poposition, using (, ( and (35 to bound K C(N,, β, p u p f q, and this ends the poof of the Poposition 5 Global decay esult Gatheing togethe Poposition and Poposition we deduce the following global esult Poposition 3 Let p, q, p > be some exponents satisfying p + q = p and p > N Let β > be any given exponent such that β < p N (38 p Then one can find an ε = ε(n, β such that the following holds Let R N be any (ε, -Reifenbeg-flat domain fo some >, and let u be a solution fo the poblem (P in with u L p ( and f L q ( Then (39 dx C +β u p f q x, (, /6, B(x, with C = C(N,, β, p Poof By Poposition and Poposition, we aleady know that (39 holds tue fo evey x, o fo points x such that dist(x, /3 It emains to conside balls centeed at points x veifying dist(x, /3 Let x be such a point Then Poposition diectly says that (39 holds fo evey adius such that < dist(x, /, and it emains to extend this fo the adii in the ange (4 dist(x, / /6

15 REGULARITY IN REIFENBERG FLAT DOMAINS 5 Fo this pupose, let y be such that dist(x, = x y /3 Denoting d(x := dist(x, we obseve that fo the that satisfies (4 we have (4 B(x, B(y, + d(x B(y, 3 Then since y, Poposition says that (4 dx C +β u p f q (, /, B(y, so that (39 follows, up to change C with 3 +β C 6 Conclusion and main esult The classical esults on Campanato Spaces can be found fo instance in [8] We define the space L p,λ ( := { u L p ( ; sup x,ρ (ρ λ u u x, p dx B(x, < + whee the supemum is taken ove all x and all ρ diam(, and whee u x, means the aveage of u on the ball B(x, A poof of the next esult can be found in [8, Theoem 3] Theoem 6 (Campanato If N < λ N + p then L p,λ ( C,α (, We can now pove ou main esult with α = λ N p Poof of Theoem Consideing u as a function of W, (R N by setting outside, and applying Poposition 3 with β = α, we obtain that (43 dx C +β u p f q x, (, /6, B(x, with C = C(N,, β, p Recalling now the classical Poincaé inequality in a ball B(x, ( u u x, dx C(N + N dx, B(x, B(x, we get (44 u u x, dx C N+ β x, (, /6, B(x, with C = C(N,, β, p, u p, f q But this implies that } Moeove β < p N p u L,N+ β (B(x, / x and hence Theoem 6 says that u C,α (B(x, /

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