Besov regularity of solutions of the p-laplace equation

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Numerical Analysis benjamin.scharf@ma.tum.

1 Besov regularity of solutions of the p-laplace equation Benjamin Scharf Technische Universität München, Department of Mathematics, Applied Numerical Analysis 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

2 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

3 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, http: // www. math. ntnu. no/ ~ lqvist/ p-laplace. pdf Benjamin Scharf Besov regularity of solutions of the p-laplace equation 3 of 23

4 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

5 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

6 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

7 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

8 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

9 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, , Benjamin Scharf Besov regularity of solutions of the p-laplace equation 8 of 23

10 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. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 9 of 23

11 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, 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

12 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 3. boundary wavelets η I,p ; (1) doesn t hold dist(i, Ω) diam(i ) (1) Benjamin Scharf Besov regularity of solutions of the p-laplace equation 10 of 23

13 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

14 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

15 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: , Benjamin Scharf Besov regularity of solutions of the p-laplace equation 12 of 23

16 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) Benjamin Scharf Besov regularity of solutions of the p-laplace equation 13 of 23

17 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): , Benjamin Scharf Besov regularity of solutions of the p-laplace equation 13 of 23

18 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

19 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, Benjamin Scharf Besov regularity of solutions of the p-laplace equation 14 of 23

20 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 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

21 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

22 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

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 = p ( ) p p (p 1) 2 > max 2, p 1 Benjamin Scharf Besov regularity of solutions of the p-laplace equation 18 of 23

24 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

25 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

26 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 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 < p 1, if 2 < p <. Benjamin Scharf Besov regularity of solutions of the p-laplace equation 21 of 23

27 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 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: v2, T. Iwaniec and J. Manfredi. Regularity of p-harmonic functions on the plane. Rev. Mat. Iberoamericana, 5(1-2):119, Benjamin Scharf Besov regularity of solutions of the p-laplace equation 22 of 23

28 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

29 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 benjamin.scharf@ma.tum.de web: Benjamin Scharf Besov regularity of solutions of the p-laplace equation 23 of 23

Nonlinear aspects of Calderón-Zygmund theory

Nonlinear aspects of Calderón-Zygmund theory Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with