Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916147744473088 |
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| author | Khan, Gabriel Saha, Soumyajit Tuerkoen, Malik |
| author_facet | Khan, Gabriel Saha, Soumyajit Tuerkoen, Malik |
| contents | We study the Dirichlet problem for the weighted Schrödinger operator \[-Δu +Vu = λρu,\] where $ρ$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations Khan, Gabriel Saha, Soumyajit Tuerkoen, Malik Differential Geometry Analysis of PDEs Spectral Theory 58J50, 53C18, 35P15 We study the Dirichlet problem for the weighted Schrödinger operator \[-Δu +Vu = λρu,\] where $ρ$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large. |
| title | Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 58J50, 53C18, 35P15 |
| url | https://arxiv.org/abs/2403.03200 |