Estimates for harmonic functions near pseudo-corners

TL;DR

This paper investigates the unbounded behavior of derivatives of harmonic functions near corners in domains, providing explicit estimates for Sobolev norms as corners are approximated or smoothed.

Contribution

It introduces a method using inner expansions and generalized power series to derive explicit Sobolev norm estimates near domain corners.

Findings

Sobolev norms blow up as corners are smoothed or approximated.

Explicit power-law estimates for singularity coefficients.

Applicable to Laplace problems with corner approximations.

Abstract

It is well known that derivatives of solutions to elliptic boundary value problems may become unbounded near the corner of a domain with a conical singularity, even if the data are smooth. When the corner domain is approximated by more regular domains, then higher order Sobolev norms of the solutions on these domains can blow up in the limit. We study this blow-up in the simple example of the Laplace operator with Dirichlet conditions in two situations: The rounding of a corner in any dimension, and the two-dimensional situation where a polygonal corner is replaced by two or more corners with smaller angles. We show how an inner expansion derived from a more general recent result on converging expansions into generalized power series can be employed to prove simple and explicit estimations for Sobolev norms and singularity coefficients by powers of the approximation scale.