Spectral analysis of hypoelliptic Višik–Ventcel’ boundary value problems

Abstract

This paper is devoted to the study of a class of hypoelliptic Visik–Ventcel’ boundary value problems for second order, uniformly elliptic differential operators. Our boundary conditions are supposed to correspond to the diffusion phenomenon along the boundary, the absorption and reflection phenomena at the boundary in probability. If the absorbing boundary portion is not a trap for Markovian particles, then we can prove two existence and uniqueness theorems of the non-homogeneous Visik–Ventcel’ boundary value problem in the framework of $$L^{2}$$ Sobolev spaces. Moreover, if the absorbing boundary portion is empty, then we can prove a generation theorem of analytic semigroups for the closed realization of the uniformly elliptic differential operator associated with the hypoelliptic Visik–Ventcel’ boundary condition in the $$L^{2}$$ topology. As a by-product, this paper is the first time to prove the angular distribution of eigenvalues, the asymptotic eigenvalue distribution and the completeness of generalized eigenfunctions of the closed realization, similar to the elliptic (non-degenerate) case.