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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