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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Fredholm property of general elliptic problems


Authors: A. Volpert and V. Volpert
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 67 (2006).
Journal: Trans. Moscow Math. Soc. 2006, 127-197
MSC (2000): Primary 35J25; Secondary 34D09, 47F05
Published electronically: December 27, 2006
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Abstract: Linear elliptic problems in bounded domains are normally solvable with a finite-dimensional kernel and a finite codimension of the image, that is, satisfy the Fredholm property, if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii condition are satisfied. In the case of unbounded domains these conditions are not sufficient any more. The necessary and sufficient conditions of normal solvability with a finite-dimensional kernel are formulated in terms of limiting problems. Adjoint operators to elliptic operators in unbounded domains are studied and the conditions in order for them to be normally solvable with a finite-dimensional kernel are also formulated by means of limiting problems. The properties of the direct and of the adjoint operators are used to prove the Fredholm property of elliptic problems in unbounded domains. Some special function spaces introduced in this work play an important role in the study of elliptic problems in unbounded domains.


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Additional Information

A. Volpert
Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel

V. Volpert
Affiliation: Laboratoire de Mathématiques Appliquées, UMR 5585 CNRS, and Université Lyon 1, 69622 Villeurbanne, France

DOI: http://dx.doi.org/10.1090/S0077-1554-06-00159-2
PII: S 0077-1554(06)00159-2
Published electronically: December 27, 2006
Dedicated: Dedicated to Ya. B. Lopatinskii on the occasion of his 100th birthday anniversary
Article copyright: © Copyright 2006 American Mathematical Society