Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids

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 $$\Omega $$ .