On the Poincaré inequality on open sets in $\mathbb{R}^n$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916099339059200 |
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| author | Gallagher, A. -K. |
| author_facet | Gallagher, A. -K. |
| contents | We show that the Poincaré inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13641 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Poincaré inequality on open sets in $\mathbb{R}^n$ Gallagher, A. -K. Analysis of PDEs Complex Variables Spectral Theory 35P15, 31B99, 32W05 We show that the Poincaré inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian. |
| title | On the Poincaré inequality on open sets in $\mathbb{R}^n$ |
| topic | Analysis of PDEs Complex Variables Spectral Theory 35P15, 31B99, 32W05 |
| url | https://arxiv.org/abs/2307.13641 |