Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids

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Abstract

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 \(\Omega \).

Keywords

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 

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Authors and Affiliations

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

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