Representation of a solution of the dirichlet problem in a planar cusp domain as a logarithmic single-layer potential

The problem of representing the solution of the Dirichlet problem for the Laplace equation as a single-layer potential V _ϱ with unknown density ϱ is known to lead to the equation V _ϱ = f for density ϱ, where f is the Dirichlet boundary data. Let Γ be the boundary of a bounded planar domain with an outward or inward peak and T(Γ) be the space of the traces on Γ of functions with finite Dirichlet integral over R ². It is shown that the operator $ L_2 left( Gamma right) ominus 1 mathrelbackepsilon varrho to Vleft. varrho right|Gamma in Tleft( Gamma right) $ L_2 left( Gamma right) ominus 1 mathrelbackepsilon varrho to Vleft. varrho right|Gamma in Tleft( Gamma right) is continuous, and the operator $ varrho to Vvarrho - overline {Vvarrho } $ varrho to Vvarrho - overline {Vvarrho } (where $ bar u $ bar u denotes u averaged over Γ) can be uniquely extended to the isomorphism