On the Torsion Function for Simply Connected, Open Sets in \(\mathbb {R}^2\)

For an open set \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ let \(\lambda (\Omega )\) $\lambda (\mathrm{\Omega })$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in \(L^2(\Omega )\) $L^2(\mathrm{\Omega })$ . Let \(w_\Omega \) $w_{\mathrm{\Omega }}$ be the torsion function for \(\Omega \) $\mathrm{\Omega }$ , and let \(\Vert .\Vert _p\) ${\Vert .\Vert }_p$ denote the \(L^p\) $L^p$ norm. It is shown that there exist \(\eta _1>0,\eta _2>0\) ${\eta }_1>0,{\eta }_2>0$ such that (i) \(\Vert w_{\Omega }\Vert _{\infty } \lambda (\Omega )\ge 1+\eta _1\) $\Vert w_{\mathrm{\Omega }}{\Vert }_{\infty }\lambda (\mathrm{\Omega })\ge 1+{\eta }_1$ for any non-empty, open, simply connected set \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with \(\lambda (\Omega ) >0\) $\lambda (\mathrm{\Omega })>0$ , (ii) \(\Vert w_{\Omega }\Vert _1\lambda (\Omega )\le {(1-\eta _2)}|\Omega |\) $\Vert w_{\mathrm{\Omega }}{\Vert }_1\lambda (\mathrm{\Omega })\le (1-{\eta }_2)|\mathrm{\Omega }|$ for any non-empty, open, simply connected set \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with finite measure \(|\Omega |\) $|\mathrm{\Omega }|$ .