2018/10/18 by Russ, Emmanuel, Trey, Baptiste, Velichkov, Bozhidar · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1810.07943
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)⋅ ∇ 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 ≥ 0, we show that the minimum of the following non-variational problem min λ_1(Ω, V) : Ω ⊂ D quasi-open, |Ω| ≤ m, |V|_∞ ≤ 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 λ_1(Ω, Φ) : Ω ⊂ D quasi-open, |Ω| ≤ m , where Φ is a given Lipschitz function on D. We prove that the topological boundary ∂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 lt; d * , discrete if d = d * and of locally finite Hd--d * Hausdorff measure if d gt; 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 C 1,α in a neighborhood of x, for some α ≤ 1 /2. This last result is optimal in the sense that C 1,1/2 is the best regularity that one can expect.