Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929283423797248 |
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| author | Leylekian, Roméo |
| author_facet | Leylekian, Roméo |
| contents | In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_15523 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate Leylekian, Roméo Analysis of PDEs Optimization and Control 35P05, 35G05, 49R05 In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains. |
| title | Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate |
| topic | Analysis of PDEs Optimization and Control 35P05, 35G05, 49R05 |
| url | https://arxiv.org/abs/2306.15523 |