Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter

This paper is motivated by the study of the existence of optimal domains maximizing the k th Robin Laplacian eigenvalue among sets of prescribed measure, in the case of a negative boundary parameter. We answer positively to this question and prove an existence result in the class of measurable sets and for quite general spectral functionals. The key tools of our analysis rely on tight isodiametric and isoperimetric geometric controls of the eigenvalues. In two dimensions of the space, under simply connectedness assumptions, further qualitative properties are obtained on the optimal sets.