Spectral analysis of the subelliptic oblique derivative problem

This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter $\lambda$. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when $\lvert \lambda \rvert$ tends to $\infty$. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of $L^2$ Sobolev spaces.

Keywords

oblique derivative problem, subelliptic operator, asymptotic eigenvalue distribution, Boutet de Monvel calculus

2010 Mathematics Subject Classification

35J25, 35P20, 35S05, 47D03

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Received 16 March 2016

Accepted 29 September 2016

Published 26 September 2017