Existence and regularity of optimal shapes for elliptic operators with drift.

This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift $L = -\Delta + V(x) \cdot \nabla$ with Dirichlet boundary conditions, where $V$ is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue $\lambda_1(\Omega,V)$ for a bounded quasi-open set $\Omega$ which enjoys similar properties to the case of open sets. Then, given $m>0$ and $\tau\geq 0$, we show that the minimum of the following non-variational problem \begin{equation*} \min\Big\{\lambda_1(\Omega,V)\ :\ \Omega\subset D\ \text{quasi-open},\ |\Omega|\leq m,\ \|V\|_{L^\infty}\le \tau\Big\}. \end{equation*} is achieved, where the box $D\subset \mathbb{R}^d$ is a bounded open set. The existence when $V$ is fixed, as well as when $V$ varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape $\Omega^\ast$ solving the minimization problem \begin{equation*} \min\Big\{\lambda_1(\Omega,\nabla\Phi)\ :\ \Omega\subset D\ \text{quasi-open},\ |\Omega|\leq m\Big\}, \end{equation*} where $\Phi$ is a given Lipschitz function on $D$. We prove that the topological boundary $\partial\Omega^\ast$ is composed of a {\it regular part} which is locally the graph of a $C^{1,\alpha}$ function and a {\it singular part} which is empty if $d d^\ast$, where $d^\ast \in \{5,6,7\}$ is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if $D$ is smooth, we prove that, for each $x\in \partial\Omega^{\ast}\cap \partial D$, $\partial\Omega^\ast$ is $C^{1,\alpha}$ in a neighborhood of $x$, for some $\alpha\leq \frac 12$. This last result is optimal in the sense that $C^{1,1/2}$ is the best regularity that one can expect.