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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, krupchyk@joyx.joensuu.fi, http://www.joensuu.fi/mathematics/department/personnel/krupchyk.htm
Jukka Tuomela
Affiliation:
Dept. of Mathematics, University of Joensuu, Finland, jukka.tuomela@joensuu.fi, 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

References

1. Adams, W. and Loustaunau, P., An introduction to Gröbner bases, Graduate Studies in Mathematics 3 (Amer. Math. Soc, Providence, RI, 1994).Google Scholar
2. Agmon, S., Douglis, A. and Nirenberg, L., ‘Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I’, Comm. Pure Appl. Math. 12 (1959) 623–727.CrossRefGoogle Scholar
3. Agmon, S., Douglis, A. and Nirenberg, L., ‘Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II’, Comm. Pure Appl. Math. 17 (1964) 35–92.CrossRefGoogle Scholar
4. Agranovich, M. S., ‘Elliptic boundary problems’, Partial differential equations IX, ed. Agranovich, M. S., Egorov, Yu. V. and Shubin, M. A., Encyclopaedia of Mathematical Sciences 79 (Springer, Berlin/Heidelberg, 1997) 1–144.CrossRefGoogle Scholar
5. Bastida, J. R., Field extensions and Galois theory, Encyclopaedia of Mathematics and its Applications 22 (Addison-Wesley Publishing Company, 1984).CrossRefGoogle Scholar
6. Belanger, J., Hausdorf, M. and Seiler, W., ‘A MuPAD library for differential equations’, Computer Algebra in Scientific Computing - CASC 2001 (ed. Ghanza, V. G., Mayr, E. W. and Vorozhtsov, E. V.; Springer, Berlin/Heidelberg, 2001) 25–42.CrossRefGoogle Scholar
7. Douglis, A. and Nirenberg, L., ‘Interior estimates for elliptic systems of partial dif ferential equations’, Comm. PureAppl. Math. 8 (1955) 503–538.CrossRefGoogle Scholar
8. Dudnikov, P. I. and Samborski, S. N., ‘Noetherian boundary value problems for overdetermined systems of partial differential equations’, Dokl. Akad. Nauk SSSR 258 (1981) 283–288 (in Russian), Soviet Math. Dokl. 23 (1981) (English translation).Google Scholar
9. Dudnikov, P. I. and Samborski, S. N., ‘Linear overdetermined systems of partial differential equations. Initial and initial-boundary value problems’, Partial differential equations VIII, (ed. Shubin, M. A.), Encyclopaedia of Mathematical Sciences 65 (Springer, Berlin/Heidelberg, 1996) 1–86.CrossRefGoogle Scholar
10. Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150 (Springer, New York, 1995).Google Scholar
11. Gerhard, J., Oevel, W., Postel, F. and Wehmeier, S., MUPAD tutorial (Springer, 2000), http://www.mupad.de/.CrossRefGoogle Scholar
12. Greuel, G.-M. and Pfister, G., Singular, A. introduction to commutative algebra (Springer, 2002).CrossRefGoogle Scholar
13. Greuel, G.-M., Pfister, G. and Schönemann, H., ‘SINGULAR 2.0. a computer algebra system for polynomial computations’, http://www.singular.uni-kl.de.Google Scholar
14. Ince, E. L., Ordinary differential equations (Dover, 1956).Google Scholar
15. Krupchyk, K., Seiler, W. and Toumela, J., ‘Overdetermined elliptic PDEs’, J. Found. Comp. Math., to appearGoogle Scholar
16. Lopatinskij, YA. B., ‘A method of reduction of boundary value problems for systems of differential equations of elliptic type to a system of regular integral equations’, Ukr. Mat. Zh. 5 (1953) 123–151.Google Scholar
17. Mohammadi, B. and Tuomela, J., ‘Simplifying numerical solution of constrained PDE systems through involutive completion’, M2ANMath. Model. Numer. Anal, to appear.Google Scholar
18. Mora, T., Solving polynomial equation system I: the Kronecker-Duval philosophy, Encyclopaedia of Mathematics and its Applications 88 (Cambridge University Press, 2003).CrossRefGoogle Scholar
19. Pommaret, J. F., Systems of partial differential equations and- Lie pseudogroups, Mathematics and its Applications 14 (Gordon and Breach Science Publishers, 1978).Google Scholar
20. Pommaret, J. F. and Quadrat, A., ‘Algebraic analysis of linear multidimensional control systems’, IMA J. Math. Control Inform. 16 (1999) 275–297.CrossRefGoogle Scholar
21. Saunders, D., The geometry of jet bundles, London Math. Soc. Lecture Note Series 142 (Cambridge University Press, 1989).CrossRefGoogle Scholar
22. Seiler, W. M., ‘Involution - the formal theory of differential equations and its applications in computer algebra and numerical analysis’, Habilitation thesis, Dept. of Mathematics, University of Mannheim, 2001 (Springer, to appear).Google Scholar
23. Shapiro, Z. YA., ‘On general boundary problems for equations of elliptic type’, Izvestiya Akad. Nauk SSSR. Ser Mat 17 (1953) 539–562 (in Russian).Google Scholar
24. Spencer, D., ‘Overdetermined systems of linear partial differential equations’, Bull. Amer. Math. Soc 75 (1969) 179–239.CrossRefGoogle Scholar
25. Tarkhanov, N. N., translated from the 1990 Russian original by P. M. Gauthier and revised by the author, Complexes of differential operators, Mathematics and its Applications 340 (Kluwer, 1995).CrossRefGoogle Scholar