Regularity of Minimizers in Nonlinear Elasticity – the Case of a One-Well Problem in Nonlinear Elasticity

Abstract

In this note sufficient conditions for bounds on the deformation gradient of a minimizer of a variational problem in nonlinear elasticity are reviewed. As a specific model class, energy densities which are the relaxation of the squared distance function to compact sets are considered and estimates in the space of functions with bounded oscillation are presented. An explicit example related to a one-well problem shows that assumptions of convexity are essential for uniform bounds on the deformation gradient. As an application of the relaxation of the energy in this special case it is indicated how general relaxation formulas for energies with p-growth can be obtained if the relaxation with quadratic growth satisfies natural assumptions.