Analytic semigroups for the subelliptic oblique derivative problem

Abstract

This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for second-order, elliptic differential operators with a complex parameter λ. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of L^p Sobolev spaces when | λ | tends to ∞. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the L^p topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter λ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.