Optimizing the ground of a Robin Laplacian: asymptotic behavior

In this note we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. By using this relation we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications.