Optimizing the Ground-State Energy of a Robin Laplacian: Asymptotic Behavior

In this paper we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain Ω by optimizing the Robin parameter function under an integral constraint. The main novelty of our approach lies in establishing a close relation between the problem under consideration and the asymptotic behavior of the Dirichlet heat content of Ω. By using this relation, we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications.