Variational characterization of principal eigenvalues for a general class of linear weighted BVP's under mixed boundary conditions of non-classical type

The main goal of this paper is to characterize the existence of principal eigenvalues of \ arrayll -div(A∇ u) +qu=λm u & in \;\; Ω,\\[1ex] B u = 0 & on \;\; ∂ Ω, array . where m∈ L^∞(Ω) satisfies some of the conditions (A1)-(A3) below, as well as to deliver their variational characterizations whenever they exist. As we are dealing with general boundary operators of mixed non-classical type, in the general setting of this paper our results are far more general than all existing results. The case when m satisfies (A2) has not been previously explored in the literature.