A symbolic calculus for layer potentials on 𝐶¹ curves and 𝐶¹ curvilinear polygons

Lewis, Jeff E.

doi:10.1090/S0002-9939-1991-1043413-4

A symbolic calculus for layer potentials on $C^ 1$ curves and $C^ 1$ curvilinear polygons
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Proc. Amer. Math. Soc. 112 (1991), 419-427 Request permission

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

A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 466568, DOI 10.1073/pnas.74.4.1324
A. P. Calderon, C. P. Calderon, E. Fabes, M. Jodeit, and N. M. Rivière, Applications of the Cauchy integral on Lipschitz curves, Bull. Amer. Math. Soc. 84 (1978), no. 2, 287–290. MR 460656, DOI 10.1090/S0002-9904-1978-14478-4
Jonathan Cohen and John Gosselin, The Dirichlet problem for the biharmonic equation in a $C^{1}$ domain in the plane, Indiana Univ. Math. J. 32 (1983), no. 5, 635–685. MR 711860, DOI 10.1512/iumj.1983.32.32044
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. (2) 116 (1982), no. 2, 361–387 (French). MR 672839, DOI 10.2307/2007065
Martin Costabel, Singular integral operators on curves with corners, Integral Equations Operator Theory 3 (1980), no. 3, 323–349. MR 580713, DOI 10.1007/BF01701497
Johannes Elschner, Asymptotics of solutions to pseudodifferential equations of Mellin type, Math. Nachr. 130 (1987), 267–305. MR 885635, DOI 10.1002/mana.19871300125

Potential techniques for boundary value problems on

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Jeff E. Lewis, Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains, Trans. Amer. Math. Soc. 320 (1990), no. 1, 53–76. MR 1005935, DOI 10.1090/S0002-9947-1990-1005935-5
Jeff E. Lewis and Cesare Parenti, Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations 8 (1983), no. 5, 477–544. MR 695401, DOI 10.1080/03605308308820276