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

Abstract

For an open set $\Om \subset \R^2$ let $λ(\Om)$ 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 $\Vert .{\Vert }_p$ denote the $L^p$ norm. It is shown that there exist {$\hat{I}{\cdot }_1>0,\hat{I}{\cdot }_2>0$} such that { (i) $\|w_{\Om}\|_{\infty} λ(\Om)\ge 1+η_1$ for any non-empty, open, simply connected set $\Om\subset \R^2$ with $\lb(\Om) >0$, (ii) $\|w_{\Om}\|_1λ(\Om)\le {(1-η_2)}|\Om|$ for any non-empty, open, simply connected set $\Om\subset\R^2$ with finite measure $|\Om|$.

19 pages, 2 figures