On the torsion function with mixed boundary conditions

TL;DR

This paper studies the asymptotic behavior of the torsion function with mixed boundary conditions in certain domains, linking it to the first eigenvalue of the Laplacian, and extends previous non-trap domain results.

Contribution

It provides precise asymptotic estimates for the torsion function with mixed boundary conditions as the domain shrinks, connecting to spectral properties of the Laplacian.

Findings

Asymptotic behavior of the torsion function as  o 0

Quantitative estimates relating torsion function to Laplacian eigenvalues

Extension of non-trap domain results to mixed boundary conditions

Abstract

Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\varepsilon\downarrow 0$ of the $L^{\infty}$ norm of the torsion function with a Neumann boundary condition on $\partial D$, and a Dirichlet boundary condition on $\partial (\varepsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that $D\setminus K$ is a non-trap domain.