A note on the shift theorem for the Laplacian in polygonal domains (extended version)

TL;DR

This paper establishes a shift theorem for solutions to the Poisson equation in polygonal domains, detailing how boundary conditions and cone angles influence the theorem's applicability, with advanced functional space descriptions at critical points.

Contribution

It extends the shift theorem to polygonal domains with various boundary conditions, incorporating Besov spaces at critical angles, thus broadening the theorem's applicability.

Findings

Shift theorem holds for solutions in polygonal domains depending on cone angles.

At critical angles, the theorem is described using Besov spaces.

The results apply to Dirichlet, Neumann, and mixed boundary conditions.

Abstract

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.