Integral Equations and Operator Theory

, Volume 12, Issue 6, pp 835–854

Corner singularity for transmission problems in three dimensions

  • Stephan Rempel


For a transmission problem for the Laplace operator the unique solvability is proved in natural Sobolev spaces in the case when edges and corners are present. The behaviour of the solution near the corner is reduced to the question when an explicitely given meromorphic family of one-dimensional integral operators on a geodesic polygon on the two sphere has a non-trivial kernel.


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  1. [B/W] A. E. Beagles, J. R. Whiteman, Treatment of a re-entrant vertex in a three-dimensional Poisson problem. In: Singularities and constructive Methods for their Treatment. Eds. P.Grisvard, W.Wendland, J.R.Whiteman, Springer LN in Math. 1121Google Scholar
  2. [C/K] D. Colton, R. Kress. Integral Equation Methods in Scattering Theory. Wiley, New York 1983Google Scholar
  3. [C/S] M. Costabel, E. Stephan. A direct Boundary Integral Method for Transmission Problems. J. Math. Anal. Appl.106 (1985) 367–413Google Scholar
  4. [D] M. Dauge, Régularités et singularités des solutions de problèmes aux limites elliptiques sur les domains singuliers de type coins. Thesis Nantes 1986Google Scholar
  5. [G 1] P. Grisvard, Problèmes aux limites dans les polygones. Mode d'emploi. E.D.F. Bull. de la Dir. des Etudes et Rech. Ser. C Math. No 1, 1986, 21–59Google Scholar
  6. [G 2] P. Grisvard, Edge behaviour of the solution of an elliptic problem. Math. Nachr.132 (1987) 281–299Google Scholar
  7. [K 1] V. A. Kondratiev, Boundary problems for elliptic equations in domains with conical points. Trudy Mosk. Mat. Ob.16 (1967) 209–292Google Scholar
  8. [K 2] C. E. Kenig, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains. Sem. Goulaouic-Meyer-Schwartz 1983–84, exp. no XXIGoogle Scholar
  9. [K/R] R. Kress, G. R. Roach, Transmission problems for the Helmholtz equation. J. Math. Phys.19 (1978) 1433–1437Google Scholar
  10. [K/S] J. R. Kuttler, V. G. Sigilitto, Eigenvalues of the Laplacian in two dimensions SIAM Rev.26 (1984) 163–193Google Scholar
  11. [M] Y. Meyer, Théorie du potentiel dans les domains Lipschitziens, d'apres G.C. Verchota. Sem. Goulaouic —1982–84, exp. noVGoogle Scholar
  12. [M/P] V. G. Mazja, B. A. Plamenevski, Lp estimates of solutions of elliptic boundary value problems in domains with edges. Trudy Mosc. Mat. Ob.37 (1978) 49–93Google Scholar
  13. [N] A. Nikishkin, Singularities of the solutions to the Dirichlet problem for a second order equation in a neighborhood of an edge. Mosc. Univ. Math. Bull.34 (1978) 53–64Google Scholar
  14. [P/S] T. v. Petersdorf, E. Stephan, Decompositions in edge and corner singularities for the solution of the Laplacian in a polyhedron. TH Darmstadt Prep. 1150 (1988)Google Scholar
  15. [P/R 1] L. Päivärinta, S. Rempel, A deconvolution problem with the kernel 1/|x| on the plane. Applicable Analysis26 (1987) 105–128Google Scholar
  16. [P/R 2] L. Päivärinta, S. Rempel, The corner behaviour of solutions to the equation\(\Delta ^{ \mp 1/2} u = f\) in two dimensions. to appearGoogle Scholar
  17. [R] S. Rempel, Elliptic pseudo-differential operators on manifolds with corners and edges. Proc. Sommerschool “Function spaces, differential operators and nonlinear analysis”, Sodankylä 1988Google Scholar
  18. [R/S 1] S. Rempel, B.-W. Schulze, Index Theory of Elliptic Operators Akademie Verlag Berlin 1982Google Scholar
  19. [R/S 2] S. Rempel, B.-W. Schulze, Asymptotics for elliptic mixed boundary problems. Pseudo-differential and Mellin operators in spaces with conormal singularity. Mathematical Research50, Akademie Verlag Berlin 1989Google Scholar
  20. [S] B.-W. Schulze, Corner Mellin operators and reduction of orders with parameters. Preprint IMath 1988Google Scholar
  21. [W] W. Wendland, Strongly elliptic boundary integral equations. The state of the art in num.analysis. Eds. A. Iserles, M. J. D. Powell, Clarendon Press Oxford 1987, 511–562Google Scholar
  22. [W/K] H. Walden, R. B. Kellogg, Numerical determination of the fundamental eigenvalue of the Laplace operator on a spherical domain. J. Engin. Math.11 (1977) 299–318Google Scholar

Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • Stephan Rempel
    • 1
  1. 1.Karl-Weierstraß-Institut für Mathematik der AdW der DDRBerlinDDR

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