On the $L^p$ norm of the torsion unction

TL;DR

This paper derives sharp bounds for the $L^p$ norm of the torsion function in terms of domain measure and principal eigenvalue, enhancing understanding of torsion problems in mathematical analysis.

Contribution

It provides new sharp bounds for the $L^p$ norm of the torsion function based on domain measure and eigenvalues, extending previous results.

Findings

Bounds are sharp for $1\,\le p\,\le 2$

Bounds relate $L^p$ norm to Lebesgue measure and principal eigenvalue

Improves understanding of torsion function behavior in PDEs

Abstract

Bounds are obtained for the $L^p$ norm of the torsion function $v_{\Omega}$, i.e. the solution of $-\Delta v=1,\, v\in H_0^1(\Omega),$ in terms of the Lebesgue measure of $\Omega$ and the principal eigenvalue $\lambda_1(\Omega)$ of the Dirichlet Laplacian acting in $L^2(\Omega)$. We show that these bounds are sharp for $1\le p\le 2$.