Besov Regularity of Solutions to the p-Poisson Equation in the Vicinity of a Vertex of a Polygonal Domain

Abstract

We study the regularity of solutions to the p-Poisson equation, $1< p < \infty$, in the vicinity of a vertex of a polygonal domain. In particular, we are interested in smoothness results in the adaptivity scale of Besov spaces $B^{\sigma _\tau }_{\tau }(L_{\tau }(\Omega ))$, $1/ \tau = \sigma _\tau /2 + 1/p$, since the regularity in this scale is known to determine the maximal approximation rate that can be achieved by adaptive and other nonlinear approximation methods. We prove that under quite mild assumptions on the right-hand side f, solutions to the p-Poisson equation possess Besov regularity $\sigma _\tau$ for all $0< \sigma _\tau < 2$. In case f vanishes in a small neighborhood of the corner, the solutions even admit arbitrary high Besov smoothness. The proofs are based on singular expansion results and continuous embeddings of intersections of Babuska–Kondratiev spaces ${\mathcal {K}}^{\ell }_{p,a}(\Omega )$ and Besov spaces $B^s_p(L_p(\Omega ))$ into the specific scale of Besov spaces we are interested in. In regard of these embeddings, we extend the existing results to the limit case $\ell \rightarrow \infty$ by showing that the Fréchet spaces $\cap _{\ell = 1}^{\infty } {\mathcal {K}}^{\ell }_{p,a}(\Omega ) \cap B^s_p(L_p(\Omega ))$ are continuously embedded into the metrizable complete topological vector space $\cap _{\sigma _\tau > 0} B^{\sigma _\tau }_{\tau }(L_{\tau }(\Omega ))$.