Local regularity of solutions of variational problems for the equilibrium configuration of an incompressible, multiphase elastic body

We consider a multiphase, incompressible, elastic body with k preferred states: whose equilibrium configuration is described in terms of a nonconvex variational problem. We pass to a suitable relaxed variational integral whose solution has the meaning of the strain tenser and also study the associated dual problem for the stresses. At first we show that the strain tenser is smooth near any point of strict J(m)(1)-quasiconvexity of the relaxed integrand. Then we use this result to get regularity of the stress tenser on the union of pure phases at least in the two-dimensional case.