Calculus of Variations and Geometric Measure Theory

A. Henrot - D. Zucco

Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

created by zucco on 04 Feb 2017
modified on 27 Mar 2018


Accepted Paper

Inserted: 4 feb 2017
Last Updated: 27 mar 2018

Journal: Annali della Scuola Normale Superiore di Pisa
Year: 2018

ArXiv: 1702.01307 PDF


Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative properties of the maximizers, passing toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with several holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the explicit solution. We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.