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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.14448 |
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Table of Contents:
- 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 $\|.\|_p$ denote the $L^p$ norm. It is shown that there exist {$η_1>0,η_2>0$} 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 $|\Om|$.