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

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 $$\min \{{\lambda }_k(\mathrm{\Omega }):\mathrm{\Omega }quasi-openset,{\int }_{\mathrm{\Omega }}{e}^{\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}2}dx\le c\}$$ has a solution.