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A Remark on Global $ W^{1, P} $ Bounds for Harmonic Functions

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A Remark on Global $ W^{1, P} $ Bounds for Harmonic Functions A REMARK ON GLOBAL W 1;p BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES NIKOS KATZOURAKIS Abstract. In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in Lp for all finite p ≥ 1. 1. Introduction Kellogg in [K] pioneered the study of the boundary behaviour of the gradient of Harmonic functions on a bounded domain. Roughly speaking, he established that in a domain of R3 near a boundary region which can represented as the graph of a planar function, the gradient of any Harmonic function is continuous up to the boundary provided that the gradient of the boundary function and of the Harmonic function are Dini continuous themselves on the boundary. The celebrated theory of Schauder estimates [GT] establishes strong relevant results for general uniformly elliptic PDEs, providing interior and global H¨olderbounds for solutions and their derivatives in terms of the H¨oldernorms of the boundary values of the solution and the right hand side of the PDE. The Schauder theory has been improved and extended by many authors, but typically for second order elliptic PDEs with boundary values of the solutions and right hand sides of the PDEs in the H¨older spaces C2,α or C1,α, in order to obtain uniform estimates for the solutions in the respective H¨olderspaces. In [GH] Gilbarg-H¨ormanderhave extended Schauder theory to include hypothe- ses of lower regularity of the boundary values of the solution, of the boundary of the domain and of the coefficients of the equations. Troianiello [T] relaxed further some conditions of Gilbarg-H¨ormander[GH]. In the paper [HS] Hile-Stanoyevitch, extending an older result of Hardy-Littlewood [HL], proved that the gradient of a Harmonic function with Lipschitz continuous boundary values is pointwise bounded up to a constant by the logarithm of a multiple of the inverse of the distance to the arXiv:1601.00190v2 [math.AP] 1 Jul 2016 boundary. However, it appears that in none of these results, even for the special case of the Laplacian, there is an explicit global bound in Lp for the gradient of Harmonic functions which have just Lipschitz boundary values and not C1,α. In this note establish the following consequence of the result of Hile-Stanoyevitch: Theorem 1. Let n ≥ 2, Ω ⊆ Rn a bounded open set with C2 boundary. Let also g : @Ω ! R with g 2 Lip(@Ω), that is g 2 C0(@Ω) and jg(x) − g(y)j Lip(g; @Ω) := sup < 1: x;y2@Ω; x6=y jx − yj 1 2 NIKOS KATZOURAKIS (1) There exists a positive function fΩ;n :Ω ! (0; 1) depending on Ω; n such that \ p 0 (1.1) fΩ;n 2 L Ω) \ C (Ω) p2[1;1) and if u 2 C2(Ω) \ C0(Ω) is the Harmonic function solving ∆u = 0; in Ω; (1.2) u = g; on @Ω; then we have the estimate (1.3) Du(x) ≤ Lip(g; @Ω) fΩ;n(x); x 2 Ω: m 1 (2) Let (g )1 ⊆ Lip(@Ω) satisfy for some C > 0 m m (1.4) Lip(g ;@Ω) + max jg j ≤ C; m 2 N: @Ω m 1 2 0 Let also (u )1 2 C (Ω) \ C (Ω) be the Harmonic functions solving ∆um = 0; in Ω; (1.5) um = gm; on @Ω: m 1 T1 1;p Then, (u )1 is strongly precompact in p=1 W (Ω) and if (1.6) gmk −! g in C0(Ω), as k ! 1, then there is a unique limit point u 2 C2(Ω) \ C0(Ω) of the subsequence mk 1 (u )1 such that along perhaps a further subsequence (1.7) umk −! u in W 1;p(Ω) 8 p ≥ 1; as k ! 1; and the limit function u solves ∆u = 0; in Ω; u = g; on @Ω: The motivation to derive the above integrability result and its consequences comes from certain recent advances in generalised solutions of nonlinear PDE and vectorial Calculus of Variations in the space L1 ([Ka4] and [Ka2, Ka3]). The vec- torial counterparts of Harmonic functions provide useful energy comparison maps since they are \stable" in Lp for all 1 < p < 1. 2. Proofs Our notation is either self-explanatory or otherwise standard as e.g. in [E], [Ka]. The starting point of our proof is the following estimate of Hile-Stanoyevitch: under the hypotheses of Theorem1, the gradient D u of a Harmonic function u 2 C2(Ω) \ C0(Ω) which solves (1.2) with g 2 Lip(@Ω) satisfies the logarithmic estimate diam(Ω) (2.1) Du(x) ≤ C(Ω; n) Lip(g; @Ω) ln ; x 2 Ω: dist(x; @Ω) for some C depending just on Ω (and the dimension). In (2.1), diam(Ω) is the diameter of the domain and dist(x; @Ω) the distance of x from the boundary: diam(Ω) := sup jx − yj : x; y 2 Ω ; dist(x; @Ω) := inf jx − zj : z 2 @Ω : GLOBAL W 1;p BOUNDS FOR HARMONIC FUNCTIONS 3 Proof of (1) of Theorem1. Fix " > 0 smaller than the diameter of Ω and consider the inner open " neighbourhood of Ω: Ω" := x 2 Ω : dist(x; @Ω) > " : It is well known that (see e.g. [GT]) 1;1 n dist(·;@Ω) 2 Wloc (R ) and (2.2) D dist(·;@Ω) = 1; a.e. on Ω: Let p 2 [1; 1). By the Co-Area formula (see e.g. [[EG], Proposition 3, p. 118]) applied to the function p n diam(Ω) 3 x 7−! χ " (x) ln 2 R Ω dist(x; @Ω) R " (where χΩ" is the characteristic function of Ω ), we have Z diam(Ω) p ln dx = Ω" dist(x; @Ω) 0 p 1 (2.3) diam(Ω) Z diam(Ω) Z ln B dist(z; @Ω) n−1 C = B dH (z)C dt " @ fdist(·;@Ω)=tg D dist(z; @Ω) A where Hn−1 is the (n − 1)-dimensional Hausdorff measure. By using (2.2), (2.3) simplifies to Z diam(Ω) p ln dx = Ω" dist(x; @Ω) ! Z diam(Ω) Z diam(Ω) p = ln dHn−1(z) dt " fdist(·;@Ω)=tg dist(z; @Ω) Further, since dist(z; @Ω) = t, for all z 2 fdist(·;@Ω) = tg, by setting Z diam(Ω) p (2.4) I";p := ln dx Ω" dist(x; @Ω) we obtain ! Z diam(Ω) Z diam(Ω)p I";p = ln dHn−1(z) dt t (2.5) " fdist(·;@Ω)=tg Z diam(Ω) diam(Ω)p = ln Hn−1 fdist(·;@Ω) = tg dt: " t As a consequence of the regularity of the boundary, standard results regarding the equivalence between the Hausdorff measure and the Minkowski content for rectifiable sets (see e.g. [[AFP], Section 2.13, Theorem 2.106]) imply that there is a C = C(Ω) such that ess sup Hn−1 fdist(·;@Ω) = tg ≤ C(Ω) 0<t<diam(Ω) 4 NIKOS KATZOURAKIS and hence the inequality (2.5) gives Z diam(Ω) diam(Ω)p (2.6) I";p ≤ C(Ω) ln dt: " t By the change of variables diam(Ω) ! := t we can rewrite the estimate (2.6) as Z diam(Ω)=" p ";p (ln !) I ≤ C(Ω) diam(Ω) 2 d! 1 ! and by enlarging perhaps the constant C(Ω), we rewrite this as Z diam(Ω)=" ln ! p (2.7) I";p ≤ C(Ω) d!: 2=p 1 ! Claim 2. We have that lim I";p ≤ C(Ω; n; p) < 1: "!0 First proof of Claim 2 (proposed by one of the referees): By using the following known property of the Gamma function Z 1 lnp x 2 dx = Γ(1 + p) 1 x we readily conclude. Second proof of Claim 2: We now give a direct argument without quoting special functions. Consider now the function ln ! g(!) := ; g : (1; 1) ! (0; 1): !2=p Since 1 − (2=p) ln ! g0(!) = ; !(2=p)+1 we have that g is strictly increasing on (1; e2=p) and strictly decreasing on (e2=p; 1). Further, note that t 7! gp(t) also enjoys the exact same monotonicity properties since s 7! sp is strictly increasing. Moreover, since e2=p ≤ 10 for all p 2 [1; 1) and by using that " 7! I";p is decreasing (in view of (2.4)), we have Z 10 p Z 1 p ";p (ln !) (ln !) lim I ≤ C(Ω) 2 dt + 2 dt "!0 1 ! 10 ! (ln !)p Z 1 (ln !)p ≤ C(Ω) 10 sup 2 + 2 dt 1<!<10 ! 10 ! (ln !)p Z 1 (ln !)p = C(Ω) 10 + dt 2 2=p 2 ! !=e 10 ! GLOBAL W 1;p BOUNDS FOR HARMONIC FUNCTIONS 5 and hence " 1 # (2=p)p X Z k+1 (ln !)p lim I";p ≤ C(Ω) 10 + dt "!0 e4=p !2 k=10 k " 1 # (2=p)p X (ln !)p ≤ C(Ω) 10 + sup 4=p 2 e k<!<k+1 ! k=10 which gives " 1 # (2=p)p X (ln k)p (2.8) lim I";p ≤ C(Ω) 10 + : "!0 e4=p k2 k=10 Now we show that the series 1 X (ln k)p S := k2 k=10 converges. Method 1: Since the sequence (ln k)p ; k = 10; 11; 12; ::: k2 is decreasing, by the Cauchy condensation test the series S converges if and only if 1 p X m (ln k) 2 A m < 1;A := : 2 k k2 m=10 Since m+1 p p −m−1 p 2 A2m+1 (m + 1) (ln 2) 2 1 1 1 m = p p −m = 1 + −! ; 2 A2m m (ln 2) 2 2 m 2 as m ! 1, by the Ratio test we have that 1 X (ln k)p = C(p) < 1 k2 k=10 P1 m since m=10 2 A2m converges.
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