Besov regularity of solutions to the p-Poisson equation. (January 2016)

• Record Type: Journal Article
• Authors: Dahlke, Stephan; Diening, Lars; Hartmann, Christoph; Scharf, Benjamin; Weimar, Markus
• Is Part Of: Nonlinear analysis. Volume 130(2015)
• Page Start: 298
• Page End: 329
• Publication Date: 2016-01
• Subjects: 35B35 -- 35J92 -- 41A25 -- 41A46 -- 46E35 -- 65M99 -- 65T60; p-Poisson equation -- Regularity of solutions -- Hölder spaces -- Besov spaces -- Nonlinear and adaptive approximation -- Wavelets; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.10.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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 B τ σ ( L τ ( 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.