Optimizing the first Robin Eigenvalue in exterior domains: the ball’s local maximizing property

This paper builds upon the work of D. Krejčiřík and V. Lotoreichik, focusing on optimizing the lowest point of the spectrum of the Laplacian in the exterior of a compact set under attractive Robin boundary conditions. We characterize the discrete spectrum of the Laplace operator under Robin boundary conditions using a harmonic Steklov eigenvalue problem in exterior domains. Assuming the lowest point of the spectrum is a discrete eigenvalue, we show that the exterior of a ball is a local maximizer among nearly spherical domains with prescribed measure in any dimension. However, generally, it is not the global maximizer of the first Robin eigenvalue under either prescribed measure or prescribed perimeter.