*Published Paper*

**Inserted:** 4 nov 2019

**Last Updated:** 2 jan 2023

**Journal:** Calc. Var. PDE

**Year:** 2019

**Abstract:**

This paper is dedicated 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

$\qquad\qquad\min\Big\{\lambda_1(\Omega,V)\ :\ \Omega\subset D\ \text{quasi-open},\

\Omega

\leq m,\ \Vert V\Vert_{L^\infty}\le \tau\Big\}.
$

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

$\qquad\qquad\min\Big\{\lambda_1(\Omega,\nabla\Phi)\ :\ \Omega\subset D\ \text{quasi-open},\

\Omega

\leq m\Big\},$

where $\Phi$ is a given Lipschitz function on $D$. We prove that the optimal set $\Omega^\ast$ is open and that its 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 $d<d^\ast$, discrete if $d=d^\ast$ and of locally finite $\mathcal{H}^{d-d^\ast}$ Hausdorff measure 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 $C^{1,1/2}$ in a neighborhood of $x$.

**Download:**