  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.

Analysis of PDEs