Existence and regularity of optimal shapes for elliptic operators with drift

This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L= - Δ + V(x) · ∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(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≥ 0 , we show that the minimum of the following non-variational problem min{λ1(Ω,V):Ω⊂Dquasi-open,|Ω|≤m,‖V‖L∞≤τ}.is achieved, where the box D⊂ Rd 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{λ1(Ω,∇Φ):Ω⊂Dquasi-open,|Ω|≤m},where Φ is a given Lipschitz function on D. We prove that the optimal set Ω ∗ is open and that its topological boundary ∂Ω ∗ is composed of a regular part, which is locally the graph of a C1,α function, and a singular part, which is empty if d< d∗, discrete if d= d∗ and of locally finite Hd-d∗ Hausdorff measure if d> d∗, where d∗∈ { 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∈ ∂Ω ∗∩ ∂D, ∂Ω ∗ is C1 , 1 / 2 in a neighborhood of x.