Journal of the Mathematical Society of Japan

Analytic semigroups for the subelliptic oblique derivative problem

Kazuaki TAIRA

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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 $\lambda$. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of $L^{p}$ Sobolev spaces when $\vert\lambda\vert$ tends to $\infty$. 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 $\lambda$ 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.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1281-1330.

Dates
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1499846527

Digital Object Identifier
doi:10.2969/jmsj/06931281

Mathematical Reviews number (MathSciNet)
MR3685045

Zentralblatt MATH identifier
1376.35035

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35S05: Pseudodifferential operators 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Keywords
oblique derivative problem subelliptic operator analytic semigroup asymptotic eigenvalue distribution Agmon's method

Citation

TAIRA, Kazuaki. Analytic semigroups for the subelliptic oblique derivative problem. J. Math. Soc. Japan 69 (2017), no. 3, 1281--1330. doi:10.2969/jmsj/06931281. https://projecteuclid.org/euclid.jmsj/1499846527


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