arXiv2019
This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = --$Δ$+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 $λ$\_1($Ω$, V) for a bounded quasi-open set $Ω$ which enjoys similar properties to the case of open sets. Then, given m > 0 and $τ$ $\ge$ 0, we show that the minimum of the following non-variational problem min $λ$\_1($Ω$, V) : $Ω$ $\subset$ D quasi-open, |$Ω$| $\le$ m, |V|\_{\infty} $\le$ $τ$. is achieved, where the box D $\subset$ 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 $Ω$ * solving the minimization problem min $λ$\_1($Ω$, $Φ$) : $Ω$ $\subset$ D quasi-open, |$Ω$| $\le$ m , where $Φ$ is a given Lipschitz function on D. We prove that the topological boundary $\partial$$Ω$ * is composed of a regular part which is locally the graph of a C ^{1,$α$} function and a singular part which is empty if d < d * , discrete if d = d * and of locally finite H^{d--d *} Hausdorff measure if d > d * , where d * $\in$ {5, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x $\in$ $\partial$$Ω$ * $\cap$ $\partial$D, $\partial$$Ω$ * is C^{ 1,$α$} in a neighborhood of x, for some $α$ $\le$ 1 /2. This last result is optimal in the sense that C ^{1,1/2} is the best regularity that one can expect.