Besov regularity of solutions of the p-laplace equation

Besov regularity of solutions of the p-laplace equation Benjamin Scharf Technische Universität München, Department of Mathematics, Applied Numerical Analysis benjamin.scharf@ma.tum.de joint work with Lars Diening (Munich), Stephan Dahlke, Christoph Hartmann, Markus Weimar (Marburg) Jena, June 27, 2014

Overview Introduction and results for the Laplace equation (p = 2) Introduction to the p-laplace Approximation in Sobolev and Besov spaces Known results for the Laplace equation (p = 2) Sobolev and local Hölder regularity of the p-laplace Sobolev regularity of the p-laplace Local Hölder regularity of the p-laplace equation Besov regularity of solutions of the p-laplace equation From Bp,p(Ω) s and C l,α γ,loc (Ω) to Bσ τ,τ (Ω) Besov regularity of the p-laplace Benjamin Scharf Besov regularity of solutions of the p-laplace equation 2 of 23

Introduction and known results Introduction to the p-laplace The p-laplace - Introduction Ω R d Lipschitz domain, d dimension, 1 < p < Inhomogeneous p-laplace equation: p u := div ( u p 2 u ) = f in Ω, u = 0 on Ω. Variational (weak) formulation: u p 2 u, v dx = Ω Ω f v dx for all v C 0 (Ω) has a unique solution u W p 1 (Ω) for f W 1 p (Ω), has model character for nonlinear problems, similar to the Laplace equation (p = 2) for linear problems nice and free introduction: P. Lindqvist. Notes on the p-laplace equation, 2006. http: // www. math. ntnu. no/ ~ lqvist/ p-laplace. pdf Benjamin Scharf Besov regularity of solutions of the p-laplace equation 3 of 23

Introduction and known results Approximation in Sobolev and Besov spaces Sobolev and Besov spaces Wp s (Ω): Sobolev space of smoothness s and integrability p on Ω Bp,p(Ω): s Besov space of smoothness s and integrability p on Ω Benjamin Scharf Besov regularity of solutions of the p-laplace equation 4 of 23

Introduction and known results Approximation in Sobolev and Besov spaces Sobolev and Besov spaces W s p (Ω): Sobolev space of smoothness s and integrability p on Ω B s p,p(ω): Besov space of smoothness s and integrability p on Ω Wavelet representation: η I,p = I 1/2 1/p η I p-normalized wavelets g Bp,p(R s d ) g = P 0 (g) + g, ηi,p ηi,p I η Ψ and P 0 (g) L p (R d ) + g, η I,p b s p,p (R d ) < Here g, η I,p b s p,p (R d ) p = I η Ψ I sp/d g, η I,p p more smoothness more decay of the wavelet coefficients Trivial embedding: B s+ε p,p (Ω) W s p (Ω) B s p,p(ω) Benjamin Scharf Besov regularity of solutions of the p-laplace equation 4 of 23

Introduction and known results Approximation in Sobolev and Besov spaces Linear and Adaptive approximation by wavelets (i) How to approximate f B s p,p(ω), Ω bounded, by wavelet basis? Linear approximation f k of f (order k: 2 kd terms): It holds f k = P 0 (g) + I 2 k η Ψ g, ηi,p ηi,p f B s p,p(ω) (or W s p (Ω)) f f k L p (Ω) 2 ks. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 5 of 23

Introduction and known results Approximation in Sobolev and Besov spaces Linear and Adaptive approximation by wavelets (ii) Adaptive approximation f k of f (order k: 2 kd terms): f D k = P 0 (g) + (I,η) D g, ηi,p ηi,p with D = 2 kd best m-term approximation: choose D to minimize f fk D L p (Ω) : take 2 kd largest wavelet coefficients! Let 1 τ = σ d + 1 p, in particular τ < 1 possible. It holds f B σ τ,τ (Ω) f f k L p (Ω) 2 kσ Besov regularity is the maximal possible convergence rate of an adaptive algorithm how much higher than Sobolev regularity? Benjamin Scharf Besov regularity of solutions of the p-laplace equation 6 of 23

Introduction and known results Approximation in Sobolev and Besov spaces Linear and Adaptive approximation by wavelets (iii) The main reason is the following computation: Theorem Let 1 τ = σ d + 1 p, x l τ and x its non-increasing rearrangement. Then x x k p k σ d x τ, where x k is the cut-off of x after the k first terms. Proof: Assume w.l.o.g. that x τ = 1. Then x (j) τ x (k) τ 1 k x τ τ = 1 k for j > k. Therefore x x k p p x x k p τ x x k τ τ k τ p τ 1 = k σ d p. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 7 of 23

Introduction and known results Known results for the Laplace equation (p = 2) Sobolev regularity for p = 2, the linear case Theorem (Jerison, Kenig 1981,1995, Theorem B) Positive: Lipschitz domain Ω R d, f L 2 (Ω). Then the solution u of u = f in Ω, u = 0 on Ω belongs to W 3/2 2 (Ω). Negative: For any s > 3/2 there exists a Lipschitz domain Ω and smooth f s.t. u with u = f in Ω, u = 0 on Ω does not belong to W s 2 (Ω). Careful! C 1 -domain Ω and f W 1/2 2 (Ω) such that u / W 3/2 2 (Ω) D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains. J. Funct. Anal. 130, 161 219, 1995. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 8 of 23

Introduction and known results Known results for the Laplace equation (p = 2) Besov regularity for p = 2 (i) Theorem (Dahlke,DeVore 97; Jerison,Kenig 95; Hansen 2013) Lipschitz domain Ω R d, f W γ 2 (Ω) for γ max ( 4 d 2d 2, 0 ). Then the solution u of u = f in Ω, u = 0 on Ω belongs to B σ τ,τ (Ω), 1 τ = σ d + 1 p, for any σ < 3 2 d d 1. Besov reg. always better than 3/2, the maximal Sobolev regularity proof by a general embedding: small global Sobolev regularity + better local (weighted) Sobolev regularity (Babuska-Kondratiev) result in better Besov regularity! S. Dahlke, R.A. DeVore. Besov regularity for elliptic boundary value problems. Comm. Partial Differential Equations, 22(1 2), 1 16, 1997. M. Hansen, n-term approximation rates and Besov regularity for elliptic PDEs on polyhedral domains, to appear in J. Found. Comp. Math. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 9 of 23

Introduction and known results Known results for the Laplace equation (p = 2) Besov regularity for p = 2 (ii) Proof Idea: extend u to R n and take its wavelet decomposition 3 parts 1. father wavelets (independent of regularity) 2. interior and exterior wavelets η I,p with dist(i, Ω) diam(i ) (1) 3. boundary wavelets η I,p ; (1) doesn t hold handle 3 parts separately 1. no problem 2. use weighted Sobolev reg.: If f L 2 (Ω), then solution u W2 2 (Ω, w), weigth w exploding at the boundary (Babuska-Kondratiev spaces) 3. use global Sobolev reg.: If f L 2 (Ω), then solution u W 3/2 2 (Ω), use counting argument: #{η I,p boundary wav., diam(i ) 2 j } 2 j(d 1) Benjamin Scharf Besov regularity of solutions of the p-laplace equation 10 of 23

Sobolev and local Hölder regularity Table of contents Introduction and results for the Laplace equation (p = 2) Introduction to the p-laplace Approximation in Sobolev and Besov spaces Known results for the Laplace equation (p = 2) Sobolev and local Hölder regularity of the p-laplace Sobolev regularity of the p-laplace Local Hölder regularity of the p-laplace equation Besov regularity of solutions of the p-laplace equation From Bp,p(Ω) s and C l,α γ,loc (Ω) to Bσ τ,τ (Ω) Besov regularity of the p-laplace Benjamin Scharf Besov regularity of solutions of the p-laplace equation 11 of 23

Sobolev and local Hölder regularity Sobolev regularity of the p-laplace Sobolev regularity of the p-laplace Theorem (Ebmeyer 2001, 2002, Savare 1998) Ω R d bounded polyhedral domain, d 2, 1 0 (2) u L q (Ω) for q < { 3/2 ε W p (Ω), if 1 p. C. Ebmeyer. Nonlinear elliptic problems with p-structure under mixed boundary value conditions in polyhedral domains. Adv. Diff. Equ., 6:873 895, 2001. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 12 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Local Hölder regularity of the homogen. p-laplace Replacement for the local (weighted) Sobolev regularity (p = 2) Theorem (Lewis 1983; Ural ceva; Evans; DiBenedetto;...) Ω R d bounded open set, d 2, 1 0 s.t. B(x 0, 64r) Ω ( 1/p max u(x) C u dx) p C r d/p, x B(x 0,r) B(x 0,32r) α max u(x) u(y) C r x,y B(x 0,r) ( 1/p u dx) p x y α. B(x 0,32r) local (weighted) Hölder regularity for homogeneous p-laplace J. Lewis. Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. Math. J., 32(6):849 858, 1983. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 13 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Local Hölder regularity of the inhomog. p-laplace We can transfer the local Hölder regularity from the homogeneous case to the inhomogeneous p-laplace equation: Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher) Ω,d,p as before. Let α = sup{α : Theorem of Lewis holds including the estimates}. Then for u with p u = f C 1,β(α) : u is locally α-hölder continuous for α < min(α, 1/(p 1)). Analog estimates hold for local Hölder-seminorm of u. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 14 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Local Hölder regularity of the inhomog. p-laplace We can transfer the local Hölder regularity from the homogeneous case to the inhomogeneous p-laplace equation: Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher) Ω,d,p as before. Let α = sup{α : Theorem of Lewis holds including the estimates}. Then for u with p u = f C 1,β(α) : u is locally α-hölder continuous for α < min(α, 1/(p 1)). Analog estimates hold for local Hölder-seminorm of u. Problem: α (0, 1] is unknown for d 3. (later: case d = 2) T. Kuusi and G. Mingione. Guide to Nonlinear Potential Estimates. Bull. Math. Sci, 4(1):1 82, 2014. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 14 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Locally weighted Hölder spaces C l,α γ,loc (Ω) 2 k 2 k 2 kγ C 1 C 1,α 1 2 γ 2 1 2 1 1 l,α γ,loc (Ω)... Hölder space, locally weighted, with l... number of derivatives α... Hölder exponent of derivatives of order l γ... growth of Hölder exp. with distance to Ω Benjamin Scharf Besov regularity of solutions of the p-laplace equation 15 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Locally weighted Hölder spaces C l,α γ,loc (Ω) (ii) Definition (Locally weighted Hölder spaces) K compact subset of Ω, δ K distance to Ω, K family of compact subsets of Ω, g C l (Ω), set g C l,α (K) := ν =l sup x,y K, x y ν g(x) ν g(y) x y α, g C 1,α := sup δ γ γ,loc (K) K g C l,α (K) <, K K C l,α γ,loc (Ω; K) = {g C l (Ω) : g C l,α γ,loc (K) < }. K shall be the set of all B(x 0, r) such that B(x 0, 64r) Ω. This definition (l = 1) is perfectly adapted to Lewis Theorem. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 16 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation Local Hölder regularity of the p-laplace Although the optimal local Hölder regularity of the solution of the p-poisson is unknown (d 3), we can estimate γ by Lewis Theorem ( ) α 1/p max u(x) u(y) C r u p dx x y α x,y B(x 0,r) B(x 0,32r) ( 1/q C r α u dx) q x y α, p q. B(x 0,32r) C r α d/q u L q (Ω) x y α. Hence, using the result of Ebmeyer we are allowed to choose u L q (Ω) for q < pd d 1, γ = α + (d 1)/p + ε for all ε > 0. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 17 of 23

Sobolev and local Hölder regularity Local Hölder regularity of the p-laplace equation The case d = 2: Hölder regularity of the p-poisson (i) Theorem (Lindgren, Lindqvist 2013; (DDHSW 2014)) Ω R 2 bounded polygonal domain, 1 < p <, f L (Ω). If p u = f, u = 0 on Ω, then u is locally α-hölder continuous for all { 1, if 1 < p 2, α < 1 p 1, if 2 < p <. Furthermore, for the same α s, it holds u C 1,α γ,loc (Ω) for γ = α + 1/p + ε. 1 The regularity p 1 is a natural bound, take v(x) = x p/(p 1). homogen. case: Iwaniec, Manfredi (1989) proved u C l,α loc (Ω) with ( ) l + α = 1 + 1 1 + 1 6 p 1 + 1 + 14 ( ) p 1 + 1 p (p 1) 2 > max 2, p 1 Benjamin Scharf Besov regularity of solutions of the p-laplace equation 18 of 23

Besov regularity of the p-laplace equation Table of contents Introduction and results for the Laplace equation (p = 2) Introduction to the p-laplace Approximation in Sobolev and Besov spaces Known results for the Laplace equation (p = 2) Sobolev and local Hölder regularity of the p-laplace Sobolev regularity of the p-laplace Local Hölder regularity of the p-laplace equation Besov regularity of solutions of the p-laplace equation From Bp,p(Ω) s and C l,α γ,loc (Ω) to Bσ τ,τ (Ω) Besov regularity of the p-laplace Benjamin Scharf Besov regularity of solutions of the p-laplace equation 19 of 23

Besov regularity of the p-laplace equation From Bp,p s l,α (Ω) and C γ,loc (Ω) to Bσ τ,τ (Ω) From B s p,p(ω) and C l,α γ,loc (Ω) to Bσ τ,τ(ω) Theorem (Dahlke, Diening, Hartmann, S., Weimar(DDHSW) 14) Ω R d bound. Lipschitz dom., d 2, s > 0, 1 < p <, α (0, 1], { l + α, if 0 < γ < l+α σ d + 1 p = ), d d 1 (l + α + 1 p γ l+α, if d + 1 p γ < l + α + 1 p, then for all { } 0 < σ < min σ d, d 1 s we have the continuous embedding and B s p,p(ω) C l,α γ,loc (Ω) Bσ τ,τ (Ω). 1 τ = σ d + 1 p If γ not too bad and local Hölder regularity l + α is higher than Sobolev regularity s, Besov regularity σ is higher than Sobolev reg.! Benjamin Scharf Besov regularity of solutions of the p-laplace equation 20 of 23

Besov regularity of the p-laplace equation Besov regularity of the p-laplace The case d = 2: Besov regularity of the p-poisson 1. By Ebmeyer s result { Bp,p 3/2 ε (Ω), if 1 < p 2, u Bp,p 1+1/p ε (Ω), if p 2, 2. Lindgren, Lindqvist: u C 1,α γ,loc (Ω), γ = α + 1/p + ε, α < { 1, if 1 < p 2, 1 p 1, if 2 < p <. 3. γ not too bad? α + 1 p + ε = γ <? l+α d + 1 p = 1+α 2 + 1 p? Yes, α < 1 4. General embedding theorem, 1 τ = σ d + 1 p, { 2, if 1 < p 2, u B σ τ,τ (Ω) for all σ < 1 + 1 p 1, if 2 < p <. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 21 of 23

Besov regularity of the p-laplace equation Besov regularity of the p-laplace Summary: Besov regularity of the p-poisson For d = 2 results on Besov regularity beat Sobolev regularity: { 2 > 3/2, if 1 1 + 1 p if 2 1 p, if p > 2, and γ not too large depending on d. This implies p (p d, ) with p d for d. E. Lindgren and P. Lindqvist. Regularity of the p-poisson equation in the plane. arxiv:1311.6795v2, 2013. T. Iwaniec and J. Manfredi. Regularity of p-harmonic functions on the plane. Rev. Mat. Iberoamericana, 5(1-2):119, 1989. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 22 of 23

Open problems Besov regularity of the p-laplace equation Besov regularity of the p-laplace d = 2, can one do better, in dependency of the angles of the boundary? Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in L q for the p-laplace (q p) bring the Fp,q s,rloc (Ω) spaces into play... work in progress... Benjamin Scharf Besov regularity of solutions of the p-laplace equation 23 of 23

Open problems Besov regularity of the p-laplace equation Besov regularity of the p-laplace d = 2, can one do better, in dependency of the angles of the boundary? Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in L q for the p-laplace (q p) bring the Fp,q s,rloc (Ω) spaces into play... work in progress... Thank you for your attention e-mail: benjamin.scharf@ma.tum.de web: http://www-m15.ma.tum.de/allgemeines/benjaminscharf Benjamin Scharf Besov regularity of solutions of the p-laplace equation 23 of 23