Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values

TL;DR

This paper investigates the regularity of solutions to the Poisson equation with various boundary conditions in polyhedral cones within a specific Besov space framework, highlighting the solutions' increased smoothness compared to Sobolev spaces, which supports adaptive numerical methods.

Contribution

It provides a detailed analysis of solution regularity in Besov spaces for the Poisson problem with different boundary conditions, demonstrating enhanced smoothness over Sobolev spaces.

Findings

Solutions are smoother in Besov spaces than in Sobolev spaces.

The regularity results justify the effectiveness of adaptive numerical schemes.

The analysis covers homogeneous and inhomogeneous boundary data.

Abstract

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^{\alpha}_{\tau,\tau}, \ \frac{1}{\tau}=\frac{\alpha}{d}+\frac{1}{p}\ $ of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale $H^s$ in all cases, which justifies the use of adaptive schemes.