Invertible harmonic mappings of unit disk onto Dini smooth Jordan domains

TL;DR

This paper extends the Rado-Choquet-Kneser theorem to harmonic mappings with Lipschitz boundary data onto Dini smooth Jordan domains, relaxing convexity constraints by imposing a boundary Jacobian condition.

Contribution

It generalizes the classical theorem to broader boundary conditions and domain shapes without requiring convexity, using approximation techniques and recent theoretical extensions.

Findings

Extended Rado-Choquet-Kneser theorem to non-convex domains

Established boundary Jacobian conditions for harmonic mappings

Utilized approximation principles for proof

Abstract

In this paper we extend Rado-Choquet-Kneser theorem for the mappings with weak homeomorphic Lipschitz boundary data and Dini's smooth boundary but without restriction on the convexity of image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Rado-Choquet-Kneser theorem by Alessandrini and Nesi \cite{ale} and it is used the approximation principle.