Besov regularity of solutions to the [formula omitted]-Poisson equation

Abstract

In this paper, we are concerned with regularity analysis for solutions to nonlinear partial differential equations. Many important practical problems are related with the p-Laplacian. Therefore, we are particularly interested in the smoothness of solutions to the p-Poisson equation. For the full range of parameters 1<p<∞ we investigate regularity estimates in the adaptivity scale Bτσ(Lτ(Ω)), 1/τ=σ/d+1/p, of Besov spaces. The maximal smoothness σ in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to p-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local Hölder with global Sobolev estimates. In particular, we prove that intersections of locally weighted Hölder spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.