Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A symbolic calculus for layer potentials on $ C\sp 1$ curves and $ C\sp 1$ curvilinear polygons


Author: Jeff E. Lewis
Journal: Proc. Amer. Math. Soc. 112 (1991), 419-427
MSC: Primary 47G30; Secondary 35S05, 45E05, 47A53, 47G10
DOI: https://doi.org/10.1090/S0002-9939-1991-1043413-4
MathSciNet review: 1043413
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A symbolic calculus for some algebras of Mellin operators on the finite interval $ J \equiv \left[ {0,1} \right]$ is developed. The algebras are ample enough to include singular integral operators and analytic double layer potentials and their adjoints on $ {C^1}$ curves and piecewise $ {C^1}$ curves with corners. Fredholmness and the index of the operators on $ {L^p}\left( J \right)$ are completely determined by the principal symbol on $ {L^p}\left( J \right),{\text{Smb}}{{\text{l}}^{1/p}}$.


References [Enhancements On Off] (What's this?)

  • [Cal] A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), 1324-1327. MR 0466568 (57:6445)
  • [CCFJR] A. P. Calderón, C. P. Calderón, E. B. Fabes, M. Jodeit, and N. M. Rivière, Applications of the Cauchy integral on Lipschitz curves, Bull. Amer. Math. Soc. 53-I (1978), 287-290. MR 0460656 (57:649)
  • [CG] J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a $ {C^1}$ domain in the plane, Indiana Univ. Math. J. 34 (1983), 635-685. MR 711860 (85b:31004)
  • [CMM] R. R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $ {L^2}$ pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361-387. MR 672839 (84m:42027)
  • [Cos] M. Costabel, Singular integral operators on curves with corners, Integral Equations Operator Theory 3 (1980), 323-349. MR 580713 (81j:45007)
  • [E] J. Elschner, Asymptotics of solutions of pseudodifferential equations of Mellin type, Math. Nachr. 130 (1987), 267-305. MR 885635 (88g:35214)
  • [FJR] E. B. Fabes, M. Jodeit, Jr., and N. M. Rivière, Potential techniques for boundary value problems on $ {C^1}$ domains, Acta Math. 141, 263-291.
  • [L] J. E. Lewis, Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains, Trans. Amer. Math. Soc. 320 (1990), 53-76. MR 1005935 (90k:35085)
  • [LP] J. E. Lewis and C. Parenti, Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations 8 (1983), 477-544. MR 695401 (86f:35185)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47G30, 35S05, 45E05, 47A53, 47G10

Retrieve articles in all journals with MSC: 47G30, 35S05, 45E05, 47A53, 47G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1043413-4
Keywords: Layer potentials, Mellin operators, nonsmooth domains, pseudodifferential operators, symbolic calculus
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society