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

Abstract

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 $$ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) $$ is continuous, and the operator $$ \varrho \to V\varrho - \overline {V\varrho } $$ (where $$ \bar u $$ denotes u averaged over Γ) can be uniquely extended to the isomorphism (the symbol 1 refers to the orthogonality to 1). Hence any function that is harmonic in R 2Γ and has finite Dirichlet integral in R 2 can be uniquely represented in the form const , where is the above isomorphism. Thus a solution of the bilateral Dirichlet problem Δu = 0 in R 2Γ, u|Γ = f, |u(x)| ≤ const for |x| → ∞, can be uniquely written as const , where ϱ is a unique solution of the equation const .