On the torsion function for simply connected, open sets in $\R^2$
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| Format: | Preprint |
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2024
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| _version_ | 1866909388383453184 |
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| author | Berg, Michiel van den Bucur, Dorin |
| author_facet | Berg, Michiel van den Bucur, Dorin |
| 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 $|\Om|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the torsion function for simply connected, open sets in $\R^2$ Berg, Michiel van den Bucur, Dorin Spectral Theory Analysis of PDEs 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 $|\Om|$. |
| title | On the torsion function for simply connected, open sets in $\R^2$ |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2402.14448 |