Analytic semigroups for the subelliptic oblique derivative problem

摘要

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 A 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.