Isoperimetric inequality with zero magnetic field in doubly connected domains
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918019073048576 |
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| author | Ghosh, Mrityunjoy Kachmar, Ayman |
| author_facet | Ghosh, Mrityunjoy Kachmar, Ayman |
| contents | We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner boundary and Neumann boundary conditions on the outer boundary, we show that this eigenvalue is maximized when the domain is an annulus, for a fixed area and magnetic flux. As consequences, we establish geometric inequalities for eigenvalues in settings with both singular and localized magnetic fields. We also propose a conjecture for a general optimality result and establish its validity for large magnetic fluxes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isoperimetric inequality with zero magnetic field in doubly connected domains Ghosh, Mrityunjoy Kachmar, Ayman Analysis of PDEs Optimization and Control Spectral Theory 35P15, 35J10, 49R05, 81Q10 We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner boundary and Neumann boundary conditions on the outer boundary, we show that this eigenvalue is maximized when the domain is an annulus, for a fixed area and magnetic flux. As consequences, we establish geometric inequalities for eigenvalues in settings with both singular and localized magnetic fields. We also propose a conjecture for a general optimality result and establish its validity for large magnetic fluxes. |
| title | Isoperimetric inequality with zero magnetic field in doubly connected domains |
| topic | Analysis of PDEs Optimization and Control Spectral Theory 35P15, 35J10, 49R05, 81Q10 |
| url | https://arxiv.org/abs/2505.08595 |