Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

Abstract

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians − Δ Θ , Ω in L 2 ( Ω ; d n x ) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω ⊂ R n , n ∈ N , n ⩾ 2 . Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems.