# A Weierstrass representation for 2D elasticity

## TL;DR

This paper introduces a Weierstrass representation for certain elastic energy functionals in 2D elasticity, enabling a more complete theoretical understanding and establishing a global uniqueness theorem for elastic maps.

## Contribution

It develops a Weierstrass representation for elastic maps and proves a global uniqueness theorem, advancing the mathematical theory of 2D elasticity.

## Key findings

- Elastic maps admit a Weierstrass representation with holomorphic functions.
- A global uniqueness theorem for elastic maps is established.
- The theory applies to a class of energy functionals including the squared distance functional.

## Abstract

We study a class of elastic energy functionals for maps between planar domains (among them the so-called squared distance functional) whose critical points (elastic maps) allow a far more complete theory than one would expect from general elasticity theory. For some of these functionals elastic maps even admit a "Weierstrass representation" in terms of holomorphic functions, reminiscent of the one for minimal surfaces. We also prove a global uniqueness theorem that does not seem to be known in other situations.