Harmonic mapping problem and affine capacity

TL;DR

This paper explores the conditions under which harmonic homeomorphisms exist between doubly connected planar domains, connecting the problem to minimal surfaces and hyperelasticity, and highlighting open questions in the field.

Contribution

It advances understanding of the harmonic mapping problem for doubly connected domains, identifying key challenges and open questions in the area.

Findings

Analysis of harmonic homeomorphisms between doubly connected domains

Connection to minimal surface theory and hyperelasticity

Identification of open problems in harmonic mapping theory

Abstract

The Harmonic Mapping Problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents considerable challenge and leads to several interesting open questions.