Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:00:22.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary value problems for elliptic pseudodifferential operators II

Published online by Cambridge University Press:  14 November 2011

Kazuaki Taira
Kazuaki Taira
Affiliation:
Department of Mathematics, Hiroshima University, Hagashi-Hiroshima 739, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to study boundary value problems for elliptic pseudodifferential operators which originate from the problem of existence of Markov processes in probability theory, generalising some results of our previous work. Our approach has a great advantage of intuitive interpretation of sufficient conditions for the unique solvability of boundary value problems in terms of Markovian motion. In fact, we prove that if a Markovian particle moves incessantly both by jumps and continuously in the state space, not being trapped in the set where no reflection phenomenon occurs, then our boundary value problem is uniquely solvable in the framework of Sobolev spaces of LP style.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 395 - 405
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bergh, J. and Löfström, J.. Interpolation Spaces, An Introduction (Berlin: Springer, 1976).CrossRefGoogle Scholar
2Bony, J.-M., Courrège, P. and Priouret, P.. Semi-groupes de Feller sur une variété à bord compacteet problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier (Grenoble) 18 (1968), 369–521.CrossRefGoogle Scholar
3Bourdaud, G.. LP-estimates for certain non-regular pseudo-differential operators. Cotnm. Partial Differential Equations 7 (1982), 1023–33.CrossRefGoogle Scholar
4Monvel, L. Boutet de. Boundary problems for pseudo-differential operators. Ada Math. 126 (1971), 11–51.Google Scholar
5Coifman, R. R. and Meyer, Y.. Au-delà des Opérateurs Pseudo-différentiels, Astérisque 57 (Paris: Soc. Math. France, 1978).Google Scholar
6Hörmander, L.. The Analysis of Linear Partial Differential Operators III (Berlin: Springer, 1985).Google Scholar
7Rempel, S. and Schulze, B.-W.. Index Theory of Elliptic Boundary Problems (Berlin: Akademie, 1982).Google Scholar
8Taira, K.. On the existence of Feller semigroups with boundary conditions. Mem. Amer. Math. Soc. 475(1992).Google Scholar
9Taira, K.. Boundary value problems for elliptic pseudo-differential operators. Proc. Amer. Math. Soc. 123(1995), 2519–28.CrossRefGoogle Scholar
10Taira, K.. Analytic Semigroups and Semilinear Initial Boundary Value Problems, London Mathematical Society Lecture Note Series 223 (London: Cambridge University Press, 1995).CrossRefGoogle Scholar
11Taylor, M.. Pseudodifferential Operators (Princeton, NJ: Princeton University Press, 1981).CrossRefGoogle Scholar
12Triebel, H.. Interpolation Theory, Function Spaces, Differential Operators (Amsterdam: North-Holland, 1978)Google Scholar