Minimization problems for the first Dirichlet Laplacian eigenvalue with volume constraint

In this presentation, we will consider a class of minimization problems for the first Dirichlet Laplacian eigenvalue with volume constraint for partitions of a given bounded domain. We will demonstrate the existence of an optimal open partition by proving the local Lipschitz continuity of the associated eigenfunctions. Our proofs rely on a weak formulation entailing the minimization of a penalized functional, where the variables are functions rather than domains. We will employ appropriate deformations, blowup techniques, and a monotonicity formula to obtain our results.