# Globally diffeomorphic $\sigma$--harmonic mappings

## TL;DR

This paper establishes conditions under which a two-dimensional divergence-structure elliptic equation solution mapping is a global diffeomorphism, ensuring invertibility and smoothness.

## Contribution

It provides necessary and sufficient boundary conditions for the global diffeomorphism property of $\sigma$-harmonic mappings in two dimensions.

## Key findings

- Characterization of boundary conditions for diffeomorphism
- Conditions ensuring invertibility of $\sigma$-harmonic mappings
- Theoretical framework for global diffeomorphism in elliptic PDEs

## Abstract

Given a two--dimensional mapping $U$ whose components solve a divergence structure elliptic equation, we give necessary and sufficient conditions on the boundary so that $U$ is a global diffeomorphism.