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The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

Published online by Cambridge University Press:  01 February 2010

Katsiaryna Krupchyk  and
Jukka Tuomela
Katsiaryna Krupchyk
Affiliation:
Dept. of Mathematics, University of Joensuu, Finland, [email protected], http://www.joensuu.fi/mathematics/department/personnel/krupchyk.htm
Jukka Tuomela
Affiliation:
Dept. of Mathematics, University of Joensuu, Finland, [email protected], http://www.joensuu.fi/mathematics/department/personnel/tuomela.htm

Abstract

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Elliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 9 , 2006 , pp. 287 - 329
Copyright
Copyright © London Mathematical Society 2006

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