Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids

作者：Qiao, Yu^*; Li, Hengguang

来源：Integral Equations and Operator Theory, 2018, 90(2): 14.

DOI：10.1007/s00020-018-2441-y

• Double layer potential operators
• Pseudodifferential operators on Lie groupoids
• Groupoid C*-algebras
• Weighted Sobolev spaces
• Mellin transform

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 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 +/- I + K are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators +/- 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.

• 出版日期2018-4
• 单位陕西师范大学