Arkiv för Matematik

Volume 55 (2017)

Number 1

Spectral analysis of the subelliptic oblique derivative problem

Pages: 243 – 270

DOI: https://dx.doi.org/10.4310/ARKIV.2017.v55.n1.a13

Author

Kazuaki Taira (Institute of Mathematics, University of Tsukuba, Japan)

Abstract

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

Received 16 March 2016

Accepted 29 September 2016

Published 26 September 2017