Existence and regularity of optimal shapes for elliptic operators with drift

Emmanuel Russ, Baptiste Trey, Bozhidar Velichkov

Published 2018-10-18Version 1

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 $d<d^\ast$, 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 $C^{1,\alpha}$ 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.