On the torsion function for simply connected, open sets in $\R^2$

Abstract

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown that there exist {$\eta_1>0,\eta_2>0$} such that { (i) $\|w_{\Om}\|_{\infty} \lambda(\Om)\ge 1+\eta_1$ for any non-empty, open, simply connected set $\Om\subset \R^2$ with $\lb(\Om) >0$, (ii) $\|w_{\Om}\|_1\lambda(\Om)\le {(1-\eta_2)}|\Om|$ for any non-empty, open, simply connected set $\Om\subset\R^2$ with finite measure $|\Om|$.