The diagram $(λ_1,μ_1)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912176811278336 |
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| author | Ftouhi, Ilias Henrot, Antoine |
| author_facet | Ftouhi, Ilias Henrot, Antoine |
| contents | In this paper, we are interested in the possible values taken by the pair $(λ_1(Ω), μ_1(Ω))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $Ω$. We prove that, without any particular assumption on the class of open sets $Ω$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product $λ_1 μ_1$. We plot the so-called Blaschke--Santaló diagram and give some conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The diagram $(λ_1,μ_1)$ Ftouhi, Ilias Henrot, Antoine Optimization and Control In this paper, we are interested in the possible values taken by the pair $(λ_1(Ω), μ_1(Ω))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $Ω$. We prove that, without any particular assumption on the class of open sets $Ω$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product $λ_1 μ_1$. We plot the so-called Blaschke--Santaló diagram and give some conjectures. |
| title | The diagram $(λ_1,μ_1)$ |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2501.02283 |