Calculus of Variations and Geometric Measure Theory

L. Cavallina - K. Funano - A. Henrot - A. Lemenant - I. Lucardesi - S. Sakaguchi

Two extremum problems for Neumann eigenvalues

created by lucardesi on 22 Dec 2023


Submitted Paper

Inserted: 22 dec 2023
Last Updated: 22 dec 2023

Year: 2023

ArXiv: 2312.13747 PDF


Neumann eigenvalues being non-decreasing with respect to domain inclusion, it makes sense to study the two shape optimization problems $\min\{\mu_k(\Omega):\Omega \mbox{ convex},\Omega \subset D, \}$ (for a given box $D$) and $\max\{\mu_k(\Omega):\Omega \mbox{ convex},\omega \subset \Omega, \}$ (for a given obstacle $\omega$). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of {\it self-domains} that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.