Convex sets can have interior hot spots
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929620566147072 |
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| author | Pont, Jaume de Dios |
| author_facet | Pont, Jaume de Dios |
| contents | The hot spots conjecture asserts that for any convex bounded domain $Ω$ in $\mathbb R^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $Ω$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of $d$. The construction is based on an extension of the conjecture from convex sets to log-concave measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex sets can have interior hot spots Pont, Jaume de Dios Analysis of PDEs Spectral Theory 35P05 (Primary) 35K05, 35J20, 58J50 (Secondary) The hot spots conjecture asserts that for any convex bounded domain $Ω$ in $\mathbb R^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $Ω$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of $d$. The construction is based on an extension of the conjecture from convex sets to log-concave measures. |
| title | Convex sets can have interior hot spots |
| topic | Analysis of PDEs Spectral Theory 35P05 (Primary) 35K05, 35J20, 58J50 (Secondary) |
| url | https://arxiv.org/abs/2412.06344 |