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A representation for the reproducing kernel for elliptic systems

Published online by Cambridge University Press:  20 January 2009

John F. Ahner  and
Eugene P. Hamilton
John F. Ahner
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee
Eugene P. Hamilton
Affiliation:
Center for Naval Analyses, Arlington, Virginia
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The kernel function method of Bergman and Schiffer (see (1) and (2)) has recently been used by Colton and Gilbert (see (3) and (4)) in connection with approximation theory and the numerical treatment of elliptic differential equations. In (5) Gilbert and Weinacht have successfully extended the kernel function method to elliptic systems of differential equations. Essential to their work is the concept of a matrix kernel satisfying the reproducing property. This reproducing kernel is defined initially as the difference of the Neumann matrix and the Dirichlet matrix. Thus actually to obtain the kernel matrix from this definition one has to solve both a Neumann problem and a Dirichlet problem. In view of this restriction, Gilbert and Weinacht derive an ingenious representation for the reproducing kernel in terms of purely geometric quantities which are obtained directly from the fundamental matrix for the differential system.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 3 , October 1979 , pp. 227 - 232
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

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