About Hölder-regularity of the convex shape minimizing λ2

Abstract

In this article, we consider the well-known following shape optimization problem: where λ2(Ω) denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in Ω ⊂ ℝ2, and |Ω| is the area of Ω. We prove, under some technical assumptions, that any optimal shape Ω* is and is not 𝒞1,α for any . We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems.