Sufficient conditions for regularity of area-preserving deformations

Abstract

We discuss some basic regularity properties of the area-preserving deformations \({u:\Omega \mapsto \mathbb{R}^2}\) that have minimal elastic energy \({\int \limits_\Omega|\nabla u|^2}\) among a suitable class of admissible vectorfields defined on a smooth, bounded domain \({\Omega\subset \mathbb{R}^2}\). Although we restrict ourselves to the quadratic stored energy function and 2-space, most of our results extend to three dimensional setting with convex stored energy function.