Besov regularity of solutions of the p-Laplace equation Benjamin Scharf Technische Universit¨ at M¨ unchen, Department of Mathematics, Applied Numerical Analysis [email protected] joint work with Lars Diening (Munich), Stephan Dahlke, Christoph Hartmann, Markus Weimar (Marburg) Strobl, June 5, 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 H¨older regularity of the p-Laplace Sobolev regularity of the p-Laplace Local H¨older regularity of the p-Laplace equation Besov regularity of solutions of the p-Laplace equation s (Ω) and C `,α (Ω) to B σ (Ω) From Bp,p τ,τ γ,loc Besov regularity of the p-Laplace Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 2 of 20 Introduction and known results – Introduction to the p-Laplace The p-Laplace - Introduction Ω ⊂ Rd Lipschitz domain, d dimension, 1 < p < ∞ Inhomogeneous p-Laplace equation: ∆p u := div |∇u|p−2 ∇u = f in Ω, Variational (weak) formulation: Z D Z E p−2 |∇u| ∇u, ∇v dx = f v dx Ω u = 0 on ∂Ω. for all v ∈ C0∞ (Ω) Ω ˚ p1 (Ω) for f ∈ W −1 has a unique solution u ∈ W p 0 (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 20 Introduction and known results – Approximation in Sobolev and Besov spaces Sobolev and Besov spaces Wps (Ω): Sobolev space of smoothness s and integrability p on Ω s (Ω): Besov space of smoothness s and integrability p on Ω Bp,p Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 4 of 20 Introduction and known results – Approximation in Sobolev and Besov spaces Sobolev and Besov spaces Wps (Ω): Sobolev space of smoothness s and integrability p on Ω s (Ω): Besov space of smoothness s and integrability p on Ω Bp,p Wavelet representation: ηI ,p = |I |1/2−1/p ηI p-normalized wavelets XX s g ∈ Bp,p (Rd ) ⇔ g = P0 (g ) + g , ηI ,p0 ηI ,p I η∈Ψ s and P0 (g ) Lp (Rd ) + g , ηI ,p0 bp,p (Rd ) < ∞ Here p X X s p (Rd ) = |I |−sp/d g , ηI ,p0 g , ηI ,p0 bp,p I η∈Ψ more smoothness ⇔ more decay of the wavelet coefficients s+ε s Trivial embedding: Bp,p (Ω) ,→ Wps (Ω) ,→ Bp,p (Ω) Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 4 of 20 Introduction and known results – Approximation in Sobolev and Besov spaces Linear and Adaptive approximation by wavelets (i) s (Ω), Ω bounded, by wavelet basis? How to approximate f ∈ Bp,p Linear approximation fk of f (order k: ∼ 2kd terms): X X g , ηI ,p0 ηI ,p fk = P0 (g ) + |I |≥2−k η∈Ψ It holds s f ∈ Bp,p (Ω) (or Wps (Ω)) ⇒ kf − fk Lp (Ω)k . 2−ksd . Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 5 of 20 Introduction and known results – Approximation in Sobolev and Besov spaces Linear and Adaptive approximation by wavelets (ii) Adaptive approximation fk of f (order k: ∼ 2kd terms): X g , ηI ,p0 ηI ,p with |D| = 2kd fkD = P0 (g ) + (I ,η)∈D best m-term approximation: choose D to minimize f − fkD Lp (Ω) : take 2kd largest wavelet coefficients! Let 1 τ = σ d + p1 , in particular τ < 1 possible. It holds σ f ∈ Bτ,τ (Ω) ⇒ kf − fk Lp (Ω)k ∼ 2−kσd 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 20 Introduction and known results – Known results for the Laplace equation (p = 2) Sobolev regularity for p = 2, the linear case Theorem (Jerison, Kenig 1995) Positive: Lipschitz domain Ω ∈ Rd , f ∈ L2 (Ω). Then the solution u of ∆u = f in Ω, u = 0 on ∂Ω 3/2 belongs to W2 (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 W2s (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 7 of 20 Introduction and known results – Known results for the Laplace equation (p = 2) Besov regularity for p = 2 Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95;Hansen2013) Lipschitz domain Ω ∈ Rd , f ∈ W2γ (Ω) for γ ≥ max the solution u of ∆u = f in Ω, σ (Ω), belongs to Bτ,τ Benjamin Scharf 1 τ = σ d 4−d 2d−2 , 0 . Then u = 0 on ∂Ω + p1 , for any σ < Besov regularity of solutions of the p-Laplace equation 3 2 · d d−1 . 8 of 20 Introduction and known results – Known results for the Laplace equation (p = 2) Besov regularity for p = 2 Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95;Hansen2013) Lipschitz domain Ω ∈ Rd , f ∈ W2γ (Ω) for γ ≥ max the solution u of ∆u = f in Ω, σ (Ω), belongs to Bτ,τ 1 τ = σ d 4−d 2d−2 , 0 . Then u = 0 on ∂Ω + p1 , 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), 116, 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 8 of 20 Sobolev and local H¨ older 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 H¨older regularity of the p-Laplace Sobolev regularity of the p-Laplace Local H¨older regularity of the p-Laplace equation Besov regularity of solutions of the p-Laplace equation s (Ω) and C `,α (Ω) to B σ (Ω) From Bp,p τ,τ γ,loc Besov regularity of the p-Laplace Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 9 of 20 Sobolev and local H¨ older regularity – Sobolev regularity of the p-Laplace Sobolev regularity of the p-Laplace Theorem (Ebmeyer 2001,2002) Ω ⊂ Rd bounded polyhedral domain, d ≥ 2, 1 < p < ∞, f ∈ Lp0 (Ω). If ∆p u = f and u = 0 on ∂Ω, then V := |∇u| p−2 2 1/2−ε ∇u ∈ W2 (Ω) for all ε > 0. Furthermore |∇u| ∈ Lq (Ω) for q < pd d −1 and u∈ ( 3/2−ε Wep (Ω), 1+1/p−ε Wp (Ω), if 1 < p ≤ 2, if p ≥ 2, e p= p > p. 1 − 2−p 2d 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 10 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation Local H¨older 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 11 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation Local H¨older regularity of the homogen. p-Laplace Replacement for the local (weighted) Sobolev regularity (p = 2) Theorem (Lewis 1983; Ural’ceva; Evans; DiBenedetto;. . .) Ω ⊂ Rd bounded open set, d ≥ 2, 1 < p < ∞. There exists α ∈ (0, 1] s.t. u with ∆p u = 0 fulfils: ∀ x0 ∈ Ω, r > 0 s.t. B(x0 , 64r ) ⊂ Ω max |∇u(x)| ≤ C x∈B(x0 ,r ) max p R − 1/p |∇u| dx ≤ C · r −d/p , B(x0 ,32r ) |∇u(x) − ∇u(y )| ≤ C · r −α x,y ∈B(x0 ,r ) R − p |∇u| dx 1/p |x − y |α . B(x0 ,32r ) ⇒ local (weighted) H¨older 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 11 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation Local H¨older regularity of the inhomog. p-Laplace We can transfer the local H¨ older 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-H¨older continuous for α < min(α∗ , 1/(p − 1)). Analog estimates hold for local H¨ older-seminorm of u. Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 12 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation Local H¨older regularity of the inhomog. p-Laplace We can transfer the local H¨ older 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-H¨older continuous for α < min(α∗ , 1/(p − 1)). Analog estimates hold for local H¨ older-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 12 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation `,α Locally weighted H¨older spaces Cγ,loc (Ω) `,α Cγ,loc (Ω). . . H¨older space, locally weighted, with 2−k ∼ 2−k 2kγ ∼1 `. . . number of derivatives k kC 1,α ∼ 1 ∼1 ∼ 2γ ∼ 2−1 α. . . H¨older exponent of derivatives of order ` γ. . . growth of H¨older exp. with distance to ∂Ω ∼ 2−1 Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 13 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation Local H¨older regularity of the p-Laplace Although the optimal local H¨ older regularity of the solution of the p-Poisson is unknown (d ≥ 3), we can estimate γ by Lewis’ Theorem max |∇u(x) − ∇u(y )| ≤ C · r x,y ∈B(x0 ,r ) −α p R − |∇u| dx 1/p |x − y |α B(x0 ,32r ) ≤ C · r −α−d/q · k∇u Lq (Ω)k · |x − y |α , p ≤ q. Hence, using the result of Ebmeyer |∇u| ∈ Lq (Ω) for q < pd , d −1 we are allowed to choose γ = α + (d − 1)/p + ε for all ε > 0. Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 14 of 20 Sobolev and local H¨ older regularity – Local H¨ older regularity of the p-Laplace equation The case d = 2: H¨older regularity of the p-Poisson Theorem (Lindgren, Lindqvist 2013; (DDHSW 2014)) Ω ⊂ R2 bounded polygonal domain, 1 < p < ∞, f ∈ L∞ (Ω). If ∆p u = f , u = 0 on ∂Ω, then u is locally α-H¨ older continuous for all ( 1, if 1 < p ≤ 2, α< 1 if 2 < p < ∞. p−1 , Furthermore, for the same α’s, it holds 1,α (Ω) for γ = α + 1/p + ε. u ∈ Cγ,loc The regularity 1 p−1 is a natural bound, take v (x) = |x|p/(p−1) . `,α (Ω) with homogen. case: Iwaniec, Manfredi (1989) proved u ∈ Cloc s ! 1 1 14 1 p `+α=1+ 1+ + 1+ + > max 2, 6 p−1 p − 1 (p − 1)2 p−1 Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 15 of 20 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 H¨older regularity of the p-Laplace Sobolev regularity of the p-Laplace Local H¨older regularity of the p-Laplace equation Besov regularity of solutions of the p-Laplace equation s (Ω) and C `,α (Ω) to B σ (Ω) From Bp,p τ,τ γ,loc Besov regularity of the p-Laplace Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 16 of 20 `,α σ s (Ω) (Ω) and Cγ,loc (Ω) to Bτ,τ Besov regularity of the p-Laplace equation – From Bp,p `,α s σ From Bp,p (Ω) and Cγ,loc (Ω) to Bτ,τ (Ω) Theorem (Dahlke, Diening, Hartmann, S., Weimar(DDHSW) ’14) Ω ⊂ Rd bound. Lipschitz dom., d ≥ 2, s > 0, 1 < p < ∞, α ∈ (0, 1], ( ` + α, σ∗ = d d−1 ` + α + if 1 p −γ , if 0<γ< `+α d + 1 p `+α d + p1 , ≤ γ < ` + α + p1 , then for all 0 < σ < min σ ∗ , d s d −1 and 1 σ 1 = + τ d p we have the continuous embedding `,α s σ Bp,p (Ω) ∩ Cγ,loc (Ω) ,→ Bτ,τ (Ω). If γ not too bad and local H¨ older regularity ` + 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 17 of 20 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 ( 3/2−ε Bp,p (Ω), u∈ 1+1/p−ε Bp,p (Ω), if 1 < p ≤ 2, if p ≥ 2, 2. Lindgren, Lindqvist: 1,α u ∈ Cγ,loc (Ω), γ = α + 1/p + ε, α < 3. γ not too bad? α + 1 p u∈ Benjamin Scharf 1 p−1 , ? `+α 1 d + p 1 σ 1 τ = d + p +ε=γ < 4. general embedding theorem, σ Bτ,τ (Ω) ( 1, ( 2, for all σ < 1+ 1 p−1 , Besov regularity of solutions of the p-Laplace equation = if if if if 1+α 2 1 < p ≤ 2, 2 < p < ∞. + p1 ? Yes, α < 1 1 < p ≤ 2, 2 < p < ∞. 18 of 20 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 ≤ 2, it holds 1 1 1 + p−1 > 1 + p if 2 < p < ∞. For d ≥ 3 the optimal α is unknown, α ≥ 1/3 is an open conj. For d ≥ 3 to beat Sobolev regularity we need ( 1 , if 1 < p < 2, α > 21 if p > 2, p, and γ not too large depending on d. This implies p ∈ (pd , ∞) with pd → ∞ 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 19 of 20 Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace Open problems 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 Lq for the p-Laplace (q 6= p) work in progress. . . Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 20 of 20 Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace Open problems 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 Lq for the p-Laplace (q 6= p) work in progress. . . Thank you for your attention e-mail: [email protected] web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 20 of 20
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