Sufficient conditions for regularity of area-preserving deformations

We discuss some basic regularity properties of the area-preserving deformations u:Ω↦R2 that have minimal elastic energy ∫Ω|∇u|2 among a suitable class of admissible vectorfields defined on a smooth, bounded domain Ω⊂R2. 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.