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.

Spectral analysis of hypoelliptic Višik–Ventcel’ boundary value problems

“…It should be emphasized that the Višik-Ventcel' boundary condition Bγu can be defined as an element of the Sobolev space H −5/2 (Γ ) (see Theorem 5.1). Then, by arguing just as in the proof of [52,Theorem 2.2] we can obtain the following generation theorem of analytic semigroups for the closed realization A associated with the Višik-Ventcel' boundary value problem (1.5):…”

“…In this way, we can prove that if conditions (2.4), (H.1) and (H.2) are satisfied, then the mapping A = (A, Bγ) defined by formula (2.5) is bijective for s > −1/2. In other words, the Višik-Ventcel' boundary value problem (1.4) In Section 9, in order to prove an existence and uniqueness theorem for the homogeneous Višik-Ventcel' boundary value problem (1.5) in the framework of Sobolev spaces when |λ| → ∞ (Theorem 2.4), we make use of a method essentially due to Agmon ([2], [29]), just as in Taira [52], [53]. This is a technique of treating a spectral parameter λ as a second order, elliptic differential operator of an extra variable y on the unit circle S, and relating the old problem to a new one with the additional variable.…”