Besov Regularity for the Neumann Problem

This paper is concerned with the regularity of the solutions to the Neumann problem in Lipschitz domains SZ contained in Rd. Especially, we consider the specific scaleB r s(L (Q)) 117 = s/d + 1/p, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show that the solution to the Neumann problem is much smoother in the specific Besov scale than in the usual LP-Sobolev scale which justifies the use of adaptive schemes. The proofs are performed by combining some recent regularity results derived by Zanger [23] with some specific properties of harmonic Besov spaces.

[1]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[2]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[3]  G. Verchota.  Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains , 1984 .

[4]  C. Kenig,et al.  Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains , 1987 .

[5]  R. DeVore,et al.  Interpolation of Besov-Spaces , 1988 .

[6]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[7]  R. DeVore,et al.  Compression of wavelet decompositions , 1992 .

[8]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[9]  Y. Meyer.  Wavelets and Operators , 1993 .

[10]  Carlos E. Kenig,et al.  Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems , 1994 .

[11]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .

[12]  Wolfgang Dahmen,et al.  Stable multiscale bases and local error estimation for elliptic problems , 1997 .

[13]  R. DeVore,et al.  Besov regularity for elliptic boundary value problems , 1997 .

[14]  P. Wojtaszczyk,et al.  A Mathematical Introduction to Wavelets: Wavelets and smoothness of functions , 1997 .

[15]  Daniel Z. Zanger.  Regularity and boundary variations for the Neumann problem , 1997 .

[16]  Stephan Dahlke,et al.  Besov regularity for second order elliptic boundary value problems with variable coefficients , 1998 .

[17]  Marius Mitrea,et al.  Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains , 1998 .

[18]  Stephan Dahlke,et al.  Besov regularity for elliptic boundary value problems in polygonal domains , 1999 .

[19]  Daniel Z. Zanger.  The Inhomogeneous Neumann Problem in Lipschitz Domains , 2000 .

[20]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[21]  Yuesheng Xu,et al.  Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms , 2003, Adv. Comput. Math..