Transmission problems for simply connected domains in the complex plane

TL;DR

This paper investigates the existence and uniqueness of transmission problems in simply connected planar domains, employing conformal maps and Hilbert transform identities to extend classical boundary value problem results.

Contribution

It introduces a novel approach linking half-plane domain problems to general simply connected domains using conformal maps and Rellich identities.

Findings

Established solvability conditions for transmission problems in weighted Lebesgue spaces.

Connected half-plane problems to general domains via conformal maps.

Extended classical boundary value problem techniques to new transmission problem settings.

Abstract

We study existence and uniqueness of a transmission problem in simply connected domains in the plane with data in weighted Lebesgue spaces by first investigating solvability results of a related novel problem associated to a homeomorphism in the real line and domains given by the upper and lower half planes. Our techniques are based on the use of conformal maps and Rellich identities for the Hilbert transform, and are motivated by previous works concerning the Dirichlet, Neumann and Zaremba problems.