2016/09/05 by Mazzoleni, Dario, Terracini, Susanna, Velichkov, Bozhidar · 2 citations
#35R35 #47A75 #49Q10 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1609.01231
In this paper we study the regularity of the optimal sets for the shape optimization problem min\λ1(Ω)+…+λk(Ω) : Ω⊂ℝd, open , |Ω|=1\, where λ1(⋅),…,λk(⋅) denote the eigenvalues of the Dirichlet Laplacian and |⋅| the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ωk^* is composed of a relatively open regular part which is locally a graph of a C1,α function and a closed singular part, which is empty if dd^*, where the natural number d^*∈[5,7] is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.