On the First Eigenvalue of the p-Laplace Operator with Robin Boundary Conditions in the Complement of a Compact Set

We consider the first eigenvalue λ₁ of the p-Laplace operator subject to Robin boundary conditions in the exterior of a compact set. We discuss the conditions for the existence of a variational λ₁, depending on the boundary parameter, the space dimension, and p. Our analysis involves the first p-harmonic Steklov eigenvalue in exterior domains. We establish properties of λ₁ for the exterior of a ball, including general inequalities, the asymptotic behavior as the boundary parameter approaches zero, and a monotonicity result with respect to a special type of domain inclusion. In two dimensions, we generalized to p≠ 2 some known shape optimization results.