Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids



We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let \(\Omega \) be a simply connected polygon in \(\mathbb {R}^2\). Denote by K the double layer potential operator on \(\Omega \) associated with the Laplace operator \(\Delta \). We show that the operator K belongs to the groupoid \(C^*\)-algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):27–54, 2013). By combining this result with general results in groupoid \(C^*\)-algebras, we prove that the operators \(\pm I + K\) are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators \(\pm I + K\) are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on \(\Omega \).


Double layer potential operators Pseudodifferential operators on Lie groupoids Groupoid \(\hbox {C}^{*}\)-algebras Weighted Sobolev spaces Mellin transform 

Mathematics Subject Classification

Primary 45E10 58H05 Secondary 47G40 47L80 47C15 45P05 


  1. 1.
    Aastrup, J., Melo, S., Monthubert, B., Schrohe, E.: Boutet de Monvel’s calculus and groupoids I. J. Noncommut. Geom. 4(3), 313–329 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ammann, B., Ionescu, A., Nistor, V.: Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math. 11, 161–206 (2006) (electronic)Google Scholar
  3. 3.
    Ammann, B., Lauter, R., Nistor, V.: Pseudodifferential operators on manifolds with a Lie structure at infinity. Ann. Math. (2) 165(3), 717–747 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Androulidakis, I., Skandalis, G.: The holonomy groupoid of a singular foliation. J. Reine Angew. Math. 626, 1–37 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Androulidakis, I., Skandalis, G.: Pseudodifferential calculus on a singular foliation. J. Noncommut. Geom. 5(1), 125–152 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Angell, T.S., Kleinman, R.E., Král, J.: Layer potentials on boundaries with corners and edges. Časopis Pěst. Mat. 113(4), 387–402 (1988)MathSciNetMATHGoogle Scholar
  7. 7.
    Băcuţă, C., Mazzucato, A., Nistor, V., Zikatanov, L.: Interface and mixed boundary value problems on \(n\)-dimensional polyhedral domains. Doc. Math. 15, 687–745 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Volume 10 of Berkeley Mathematics Lecture Notes. American Mathematical Society, Providence (1999)Google Scholar
  9. 9.
    Carvalho,C., Nistor, V., Qiao, Y.: Fredholm conditions on non-compact manifolds: theory and examples. arXiv:1703.07953
  10. 10.
    Carvalho, C., Nistor, V., Qiao, Y.: Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electron. Res. Announc. Math. Sci. 24, 68–77 (2017)MathSciNetGoogle Scholar
  11. 11.
    Carvalho, C., Qiao, Y.: Layer potentials \(C^*\)-algebras of domains with conical points. Cent. Eur. J. Math. 11(1), 27–54 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Costabel, M.: Boundary integral operators on curved polygons. Ann. Mat. Pura Appl. 4(133), 305–326 (1983)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Courant, R., Hilbert, D.: Methods of mathematical physics, vol. II. Wiley Classics Library. Wiley, New York (1989). Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience PublicationGoogle Scholar
  15. 15.
    Debord, C., Skandalis, G.: Adiabatic groupoid, crossed product by \(\mathbb{R}_{+}^{\ast }\) and pseudodifferential calculus. Adv. Math. 257, 66–91 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Debord, C., Skandalis, G.: Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions. Bull. Sci. Math. 139(7), 750–776 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Egorov, Y., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications. Volume 93 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1997)Google Scholar
  18. 18.
    Elschner, J.: The double layer potential operator over polyhedral domains. I. Solvability in weighted Sobolev spaces. Appl. Anal. 45(1–4), 117–134 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fabes, E., Jodeit, M., Lewis, J.: Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 26(1), 95–114 (1977)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fabes, E., Jodeit, M., Rivière, N.: Potential techniques for boundary value problems on \(C^{1}\)-domains. Acta Math. 141(3–4), 165–186 (1978)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Folland, G.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)MATHGoogle Scholar
  22. 22.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
  23. 23.
    Hsiao, G., Wendland, W.L.: Boundary Integral Equations. Volume 164 of Applied Mathematical Sciences. Springer, Berlin (2008)Google Scholar
  24. 24.
    Jerison, D., Kenig, C.: The Dirichlet problem in nonsmooth domains. Ann. Math. (2) 113(2), 367–382 (1981)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jerison, D., Kenig, C.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N.S.) 4(2), 203–207 (1981)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kapanadze, D., Schulze, B.-W.: Boundary-contact problems for domains with conical singularities. J. Differ. Equ. 217(2), 456–500 (2005)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kellogg, R.: Singularities in interface problems. In: Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proceedings of Symposia, University of Maryland, College Park, Md., 1970), pp. 351–400. Academic Press, New York (1971)Google Scholar
  28. 28.
    Kenig, C.: Recent progress on boundary value problems on Lipschitz domains. In: Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984). Volume 43 of Proceedings of Symposia in Pure Mathematics, pp. 175–205. American Mathematical Society, Providence (1985)Google Scholar
  29. 29.
    Kohr, M., Pintea, C., Wendland, W.: On mapping properties of layer potential operators for Brinkman equations on Lipschitz domains in Riemannian manifolds. Mathematica 52(75), 31–46 (2010)MathSciNetMATHGoogle Scholar
  30. 30.
    Kondrat’ev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 209–292 (1967)MathSciNetGoogle Scholar
  31. 31.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Volume 85 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)Google Scholar
  32. 32.
    Kress, R.: Linear Integral Equations, 2nd edn. Volume 82 of Applied Mathematical Sciences. Springer, New York (1999)Google Scholar
  33. 33.
    Lauter, R., Monthubert, B., Nistor, V.: Pseudodifferential analysis on continuous family groupoids. Doc. Math. 5, 625–655 (2000). (electronic)MathSciNetMATHGoogle Scholar
  34. 34.
    Lauter, R., Nistor, V.: Analysis of geometric operators on open manifolds: a groupoid approach. In: Quantization of Singular Symplectic Quotients. Volume 198 of Progress in Mathematics, pp. 181–229. Birkhäuser, Basel (2001)Google Scholar
  35. 35.
    Lewis, J., Parenti, C.: Pseudodifferential operators of Mellin type. Commun. Partial Differ. Equ. 8(5), 477–544 (1983)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Li, H., Mazzucato, A., Nistor, V.: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Electron. Trans. Numer. Anal. 37, 41–69 (2010)MathSciNetMATHGoogle Scholar
  37. 37.
    Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry. Volume 124 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1987)Google Scholar
  38. 38.
    Maz’ya, V.: Boundary integral equations. In: Analysis, IV. Volume 27 of Encyclopaedia of Mathematical Sciences, pp. 127–222. Springer, Berlin (1991)Google Scholar
  39. 39.
    Maz’ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains. Volume 162 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2010)Google Scholar
  40. 40.
    Mazzeo, R., Melrose, R.: Pseudodifferential operators on manifolds with fibred boundaries. Asian J. Math. 2(4), 833–866 (1998). Mikio Sato: a great Japanese mathematician of the twentieth centuryGoogle Scholar
  41. 41.
    Mazzucato, A., Nistor, V.: Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal. 195(1), 25–73 (2010)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  43. 43.
    Medková, D.: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslov. Math. J. 47(4), 651–679 (1997)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Melrose, R.: Transformation of boundary problems. Acta Math. 147(3–4), 149–236 (1981)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Melrose, R.: The Atiyah–Patodi–Singer Index Theorem. Volume 4 of Research Notes in Mathematics. A K Peters Ltd., Wellesley (1993)Google Scholar
  46. 46.
    Melrose, R.: Geometric Scattering Theory, Stanford Lectures. Cambridge University Press, Cambridge (1995)Google Scholar
  47. 47.
    Mitrea, D., Mitrea, I.: On the Besov regularity of conformal maps and layer potentials on nonsmooth domains. J. Funct. Anal. 201(2), 380–429 (2003)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Mitrea, I.: On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons. J. Fourier Anal. Appl. 8(5), 443–487 (2002)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Am. Math. Soc. 359(9), 4143–4182 (2007) (electronic)Google Scholar
  50. 50.
    Mitrea, M., Nistor, V.: Boundary value problems and layer potentials on manifolds with cylindrical ends. Czechoslov. Math. J. 57(4), 1151–1197 (2007)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163(2), 181–251 (1999)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: \(L^P\) Hardy, and Hölder space results. Commun. Anal. Geom. 9(2), 369–421 (2001)CrossRefMATHGoogle Scholar
  53. 53.
    Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids. Volume 91 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2003)Google Scholar
  54. 54.
    Monthubert, B.: Groupoids and pseudodifferential calculus on manifolds with corners. J. Funct. Anal. 199, 243–286 (2003)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Monthubert, B., Pierrot, F.: Indice analytique et groupoïdes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 325(2), 193–198 (1997)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Nistor, V.: Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains. In: Booss, B., Grubb, G., Wojciechowski, K.P. (eds.) Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds. Volume 366 of Contemporary Mathematics, pp. 307–328. American Mathematical Society, Rhode Island (2005)Google Scholar
  57. 57.
    Nistor, V.: Desingularization of Lie groupoids and pseudodifferential operators on singular spaces. [math.DG], to appear in Comm. Anal. Geom.
  58. 58.
    Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on differential groupoids. Pac. J. Math. 189(1), 117–152 (1999)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Perfekt, K.-M., Putinar, M.: Spectral bounds for the Neumann–Poincaré operator on planar domains with corners. J. Anal. Math. 124, 39–57 (2014)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Perfekt, K.-M., Putinar, M.: The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Arch. Ration. Mech. Anal. 223(2), 1019–1033 (2017)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Qiao, Y., Nistor, V.: Single and double layer potentials on domains with conical points I: straight cones. Integral Equ. Oper. Theory 72(3), 419–448 (2012)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Renault, J.: A Groupoid Approach to \(C^{\ast } \)-Algebras. Volume 793 of Lecture Notes in Mathematics. Springer, Berlin (1980)Google Scholar
  63. 63.
    Sauter, S., Schwab, C.: Boundary Element Methods. Volume 39 of Springer Series in Computational Mathematics. Springer, Berlin (2011). Translated and expanded from the 2004 German originalGoogle Scholar
  64. 64.
    Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. I. In: Pseudo-Differential Calculus and Mathematical Physics. Volume 5 of Mathematical Topics, pp. 97–209. Akademie Verlag, Berlin (1994)Google Scholar
  65. 65.
    Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. II. In: Boundary Value Problems, Schrödinger Operators, Deformation Quantization. Volume 8 of Mathematical Topics, pp. 70–205. Akademie Verlag, Berlin (1995)Google Scholar
  66. 66.
    Schulze, B.-W.: Boundary value problems and singular pseudo-differential operators. Pure and Applied Mathematics (New York). Wiley, Chichester (1998)Google Scholar
  67. 67.
    Schulze, B.-W., Sternin, B. Shatalov, V.: Differential Equations on Singular Manifolds. Volume 15 of Mathematical Topics. Wiley-VCH Verlag Berlin GmbH, Berlin (1998). Semiclassical theory and operator algebrasGoogle Scholar
  68. 68.
    So, B.K.: On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth. Adv. Math. 237, 1–32 (2013)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Taylor, M.: Partial Differential Equations. II. Volume 116 of Applied Mathematical Sciences. Springer, New York (1996). Qualitative studies of linear equationsGoogle Scholar
  70. 70.
    Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Verchota, G., Vogel, A.: The multidirectional Neumann problem in \(\mathbb{R}^{4}\). Math. Ann. 335(3), 571–644 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA

Personalised recommendations