Double layer potentials on three-dimensional wedges and pseudodifferential operators on Lie groupoids
The paper aims to examine the analytic properties of \(\frac{1}{2}\pm K\) operator on a three-dimensional wedge \(\mathcal{W}\), where \(K\) denotes the so-called direct value of the double-layer potential, associated to the Laplacian, calculated over the wedge \(\mathcal{W}\). It is shown that \(\frac{1}{2}\pm K\) are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on \(\mathcal{W}\). Besides, the connection between the operator \(K\) with a pseudo-differential operator algebra on some Lie groupoid is investigated. The result can be used to prove the Fredholmness of \(\frac{1}{2}\pm K_\Omega\), where \(\Omega\) is ``a domain with edge singularities'' and \(K_\Omega\) is the direct value of the double layer potential operator associated to the Laplacian, calculated over \(\Omega\).
scientific article; zbMATH DE number 6877134
scientific article