Regularity results for equilibria in a variational model for fracture

In recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the form G(K,u) = integral(Omega/K) F(del u) dx + lambda integral(Omega/K) \u - g\(q) dx + beta HN-1(K boolean AND Omega), where Omega is an open, bounded subset of R-N, q greater than or equal to 1, g is an element of L-infinity(Omega;R-N),lambda,beta > 0, the bulk energy density F is quasiconvex, K subset of R-N is closed, and the admissible deformation u:Omega --> R-N is C-1 in Omega\K. One of the main issues has to do with regularity properties of the 'crack site' K for a minimising pair (K, u). In the scalar case, i.e. when u:Omega -->R, similar models were adopted to image segmentation problems, and the regularity of the 'edge' set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of G corresponding to polyconvex bulk energy densities of the form F(xi) = 1/2\xi(2) + h(det xi), where the convex function h grows linearly at infinity.