On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917583245017088 |
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| author | Farrington, Sam |
| author_facet | Farrington, Sam |
| contents | We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension $d\geq 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension $d=2$. We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12245 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian Farrington, Sam Spectral Theory Analysis of PDEs 49Q10, 49R05, 35J25, 35P15 We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension $d\geq 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension $d=2$. We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques. |
| title | On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian |
| topic | Spectral Theory Analysis of PDEs 49Q10, 49R05, 35J25, 35P15 |
| url | https://arxiv.org/abs/2308.12245 |