Spectral analysis of the hypoelliptic Robin problem

Abstract

This paper is devoted to a functional analytic approach to the study of the hypoelliptic Robin problem for a second-order, uniformly elliptic differential operator with a complex parameter $$\lambda $$ , under the probabilistic condition that either the absorption phenomenon or the reflection phenomenon occurs at each point of the boundary. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous Robin problem when $$\vert \lambda \vert $$ tends to $$\infty $$ . More precisely, we prove the spectral properties of the closed realization of the uniformly elliptic differential operator, similar to the elliptic (non-degenerate) case. However, in the degenerate case we cannot use Green’s formula to characterize the adjoint operator of the closed realization. Hence, we shift our attention to its resolvent. In the proof, we make use of the Boutet de Monvel calculus in order to study the resolvent and its adjoint in the framework of $$L^{2}$$ Sobolev spaces.