Calculus of Variations and Geometric Measure Theory

B. Bogosel - A. Henrot - M. Michetti

Optimization of Neumann Eigenvalues under convexity and geometric constraints

created by michetti1 on 06 Feb 2024



Inserted: 6 feb 2024
Last Updated: 6 feb 2024

Year: 2024


In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index $k$ for which it can be or cannot be extremal. We give some numerical examples for small values of $k$ that lead us to state some conjectures.

Keywords: shape optimization, Neumann eigenvalues