On the commutative properties of boundary integral operators

Author:
G. F. Roach

Journal:
Proc. Amer. Math. Soc. **73** (1979), 219-227

MSC:
Primary 35C15; Secondary 35J05, 45E05

DOI:
https://doi.org/10.1090/S0002-9939-1979-0516468-5

MathSciNet review:
516468

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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.

**[1]**R. A. Adams and G. F. Roach,*An intrinsic approach to radiation conditions*, J. Math. Anal. Appl. (2)**39**(1972), 433-444. MR**0308576 (46:7690)****[2]**J. L. Howland,*Symmeterising kernels and the integral equation of the first kind of classical potential theory*, Proc. Amer. Math. Soc.**19**(1968), 1-7. MR**0220956 (36:4008)****[3]**G. C. Hsaio and R. C. McCamy,*Solution of boundary value problems by integral equations of the first kind*, SIAM Rev.**15**(1973), 687-705. MR**0324242 (48:2594)****[4]**G. C. Hsaio and G. F. Roach,*On the relationship between boundary value problems*, J. Math. Anal. Appl. (to appear). MR**533513 (81b:35023)****[5]**G. C. Hsaio and W. Wendland,*On Galerkin's method for a class of integral equations of the first kind*, Applicable Anal.**6**(1977), 155-157.**[6]**R. E. Kleinman and G. F. Roach,*Boundary integral equations for the three dimensional Helmholtz equation*, SIAM Rev.**16**(1974), 214-236. MR**0380087 (52:988)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0516468-5

Keywords:
Boundary integral operators

Article copyright:
© Copyright 1979
American Mathematical Society