## Journal of the Mathematical Society of Japan

### Analytic semigroups for the subelliptic oblique derivative problem

Kazuaki TAIRA

#### 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

https://projecteuclid.org/euclid.jmsj/1499846527

Digital Object Identifier
doi:10.2969/jmsj/06931281

Mathematical Reviews number (MathSciNet)
MR3685045

Zentralblatt MATH identifier
1376.35035

#### 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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