Minimizing Eigenvalues of the Fractional Laplacian
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918215513276416 |
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| author | Zahl, Alvis |
| author_facet | Zahl, Alvis |
| contents | We study the minimizers of \begin{equation} λ_k^s(A) + |A| \end{equation} where $λ^s_k(A)$ is the $k$-th Dirichlet eigenvalue of the fractional Laplacian on $A$. Unlike in the case of the Laplacian, the free boundary of minimizers exhibit distinct global behavior. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where $λ_k$ is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimizing Eigenvalues of the Fractional Laplacian Zahl, Alvis Analysis of PDEs 35B65, 35R35 We study the minimizers of \begin{equation} λ_k^s(A) + |A| \end{equation} where $λ^s_k(A)$ is the $k$-th Dirichlet eigenvalue of the fractional Laplacian on $A$. Unlike in the case of the Laplacian, the free boundary of minimizers exhibit distinct global behavior. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where $λ_k$ is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers. |
| title | Minimizing Eigenvalues of the Fractional Laplacian |
| topic | Analysis of PDEs 35B65, 35R35 |
| url | https://arxiv.org/abs/2504.09840 |