Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

TL;DR

This paper proves the existence of minimizers for eigenvalues of a drifted Dirichlet-Laplacian operator under a specific measure constraint, advancing understanding of eigenvalue optimization in weighted domains.

Contribution

It establishes the existence of minimizers for eigenvalues of a drifted Laplacian under a measure constraint, a novel result in spectral optimization.

Findings

Existence of minimizers for eigenvalues under measure constraints

Applicable to quasi-open sets in spectral optimization

Provides a foundation for further eigenvalue shape analysis

Abstract

This paper deals with the eigenvalue problem for the operator $L=-\Delta -x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $\lambda_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N} $ the following minimization problem $$ \min\left\{\lambda_k(\Omega): \> \Omega \>\mbox{quasi-open} \>\mbox{set}, \> \int_\Omega e^{|x|^2/2}dx\le c\right\} $$ has a solution.