Attainment results for nematic elastomers

TL;DR

This paper investigates the attainment of energy minimizers in nematic elastomers, providing explicit solutions for linear models and employing convex integration techniques for nonlinear cases.

Contribution

It introduces explicit minimizers for the linearized model and applies convex integration to establish attainment results for nonlinear nematic elastomer energies.

Findings

Explicit minimizer for the linearized problem with boundary conditions

Attainment results for nonlinear models using convex integration

Extension of convex integration methods to nematic elastomer energies

Abstract

We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by M\"uller and Sver\'ak in the context of crystalline solids.