Optimal domains for elliptic eigenvalue problems with rough coefficients

TL;DR

This paper establishes the existence of optimal open sets minimizing the first Dirichlet eigenvalue for elliptic operators with rough coefficients, using a free boundary approach and regularity estimates.

Contribution

It introduces a free boundary method to identify optimal domains for elliptic eigenvalue problems with measurable coefficients, including regularity results.

Findings

Existence of an open set minimizing the eigenvalue for given measure.

Optimal eigenfunction is Lipschitz continuous at free boundary points.

Eigenfunction grows at most linearly from the free boundary.

Abstract

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a H\"older estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e. it is Lipschitz continuous at free boundary points.