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Elliptic Boundary Problems

  • Chapter
Partial Differential Equations IX

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 79))

Abstract

This paper, somewhat delayed in the series “Fundamental Directions,” is devoted to general linear elliptic boundary problems on a smooth compact manifold with boundary. The paper is intended for a large circle of readers. We hope the paper will be useful to many mathematicians with diverse scientific interests. The main features of the theory we will discuss were formed in the 60’s, beginning with the proof of the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces. This was done on the basis of achievements of many mathematicians during the preceding decades. This basis was very extensive, but the results were incomplete. The elaboration of the general theory was stimulated by investigations of the index problem (see e.g. (Palais 1965) and (Fedosov 1990)) and went on under a strong influence of the microlocal analysis, beginning with the appearance of the calculus of pseudodifferential operators. During the last decades, new variants of the general theory appeared, and the old variants were enriched by new results.

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M. S. Agranovich Yu. V. Egorov M. A. Shubin

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Agranovich, M.S. (1997). Elliptic Boundary Problems. In: Agranovich, M.S., Egorov, Y.V., Shubin, M.A. (eds) Partial Differential Equations IX. Encyclopaedia of Mathematical Sciences, vol 79. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06721-5_1

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