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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 \(\Omega \).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, Shaanxi, China

    Yu Qiao

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Hengguang Li

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  1. Yu Qiao
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  2. Hengguang Li
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Correspondence to Yu Qiao.

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Qiao was partially supported by the NSFC Grant 11301317 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Li was partially supported by the NSF Grant DMS-1418853, by the Natural Science Foundation of China (NSFC) Grant 11628104, and by the AMS Fan China Exchange Program.

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Qiao, Y., Li, H. Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids. Integr. Equ. Oper. Theory 90, 14 (2018). https://doi.org/10.1007/s00020-018-2441-y

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