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

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 $1\le p\le 2$.

14 pages