Hyperelastic deformations of smallest total energy

Throughout this article X and Y will be nonempty bounded domains in, ℝⁿ n ≧ 2. The term deformation of X ⊂ ℝⁿ onto Y ⊂ ℝⁿ refers to an orientation preserving homeomorphism h: X → Y in the Sobolev class W^1,1(X,Y) whose inverse f: Y→X lies in W^1,1(Y,X). The general law of hyperelasticity asserts that there exists an energy integral such that the elastic deformations have the smallest energy. We assume here that E is conformally coerced and polyconvex. Some additional regularity conditions are also imposed. Under those conditions we establish the existence and global invertibility of the minimizers. The key tools in obtaining an extremal deformation h: X → Y, regardless of its boundary values, are the free Lagrangians. Finding suitable free Lagrangians and using them for a specific stored-energy function E is truly a work of art. We have done it here for the so-called total harmonic energy and a pair of annuli in the plane. In fact this challenging problem illustrates rather clearly the strength of the concept of free Lagrangians.