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Local regularity of solutions of variational problems for the equilibrium configuration of an incompressible, multiphase elastic body
Michael Bildhauer, Martin Fuchs and Gregory A. Seregin
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 tensor and also study the associated dual problem for the stresses. At first we show that the strain tensor is smooth near any point of strict J1m-quasiconvexity of the relaxed integrand. Then we use this result to get regularity of the stress tensor on the union of pure phases at least in the two-dimensional case.