تقرير
On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian
| العنوان: | On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian |
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| المؤلفون: | Farrington, Sam |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 49Q10, 49R05, 35J25, 35P15 |
| الوصف: | 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. Comment: 30 pages, 3 figures. Version 2 includes shortening and corrections to some of the proofs and a new presentation of the material with some extra results that the author has become aware of |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2308.12245 |
| رقم الأكسشن: | edsarx.2308.12245 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |