Sharp bounds on the failure of the hot spots conjecture
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912549136498688 |
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| author | Pont, Jaume de Dios Hsu, Alexander W. Taylor, Mitchell A. |
| author_facet | Pont, Jaume de Dios Hsu, Alexander W. Taylor, Mitchell A. |
| contents | The hot spots ratio of a domain $Ω\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $Ω\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp bounds on the failure of the hot spots conjecture Pont, Jaume de Dios Hsu, Alexander W. Taylor, Mitchell A. Spectral Theory Mathematical Physics Analysis of PDEs The hot spots ratio of a domain $Ω\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $Ω\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$. |
| title | Sharp bounds on the failure of the hot spots conjecture |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2508.16321 |