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6 - Lp Approach to Elliptic Boundary Value Problems

Published online by Cambridge University Press:  05 April 2016

Kazuaki Taira
Affiliation:
Waseda University, Japan
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Summary

First, we formulate the degenerate elliptic boundary value problem in terms of function spaces. By using the Neumann problem, we prove that our boundary value problem can be reduced to the study of a pseudo-differential operator on the boundary. This reduction to the boundary approach is a generalization of the classical Fredholm integral equation. The virtue of this reduction is that there is no difficulty in taking adjoints after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints. This allows us to discuss the existence theory more easily. Several recent developments in the theory of pseudo-differential operators have made possible further progress in the study of elliptic boundary value problems and hence the study of semilinear parabolic initial boundary value problems. Finally, we establish the Lopatinski--Shapiro ellipticity condition for boundary value problems in the framework of vector bundles over a compact smooth Riemannian manifold with boundary, and state the most fundamental theorem for elliptic boundary value problems for differential geometry.
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Publisher: Cambridge University Press
Print publication year: 2016

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