On the commutative properties of boundary integral operators

Roach, G. F.

doi:10.1090/S0002-9939-1979-0516468-5

On the commutative properties of boundary integral operators
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Proc. Amer. Math. Soc. 73 (1979), 219-227 Request permission

Abstract:

A discussion of the interior Dirichlet and Neumann problems of classical potential theory can be given in terms of the symmeterisers of certain related integral operators. Recent developments in the theory and application of integral equations of the first kind have made this approach towards the solution of boundary value problems a more attractive proposition. However for problems more general than those arising in potential theory a greater knowledge of associated spectral properties is required together with a realisation that much of the symmetry occurring in potential problems will be lost and that attention must be directed instead towards commutativity relations. This is demonstrated by considering boundary value problems associated with the Helmholtz equation.

References

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On Galerkin’s method for a class of integral equations of the first kind

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R. E. Kleinman and G. F. Roach, Boundary integral equations for the three-dimensional Helmholtz equation, SIAM Rev. 16 (1974), 214–236. MR 380087, DOI 10.1137/1016029