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 𝒞^1,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.

Keywords:

• shape optimization
• eigenvalues of the Laplacian
• regularity of free boundaries
• conformal map
• convex constraint
• overdetermined boundary value problems

AMS Subject Classifications::

• 49Q10
• 32H40
• 49N60
• 52A10
• 35P15