تقرير
Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem
| العنوان: | Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem |
|---|---|
| المؤلفون: | Ferreri, Lorenzo, Mazzoleni, Dario, Pellacci, Benedetta, Verzini, Gianmaria |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $\Omega\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial \Omega$, tends to a point of maximal mean curvature of $\partial \Omega$. Furthermore, we show that $D$ is the intersection with $\Omega$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$. These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework. Comment: 29 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.17931 |
| رقم الأكسشن: | edsarx.2407.17931 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |