Regularity results for equilibria in a variational model for fracture
journal contributionposted on 01.01.1996 by Emilio Acerbi, Irene Fonseca, Nicola Fusco
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Abstract: "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 [equation] where [omega] is an open, bounded subset of R[superscript N], q [> or =] 1, g [element of]L[superscript infinity]([omega];R[superscript N]), [lambda], ╬▓ > 0, the bulk energy density F is quasiconvex, K [contained as subclass within] R[superscript N] is closed, and the admissible deformation u : [omega] -> R[superscript N] is C┬╣ in [omega]\K. One of the main issues has to do with regularity properties of the 'crack site' K for a minimizing 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 minimizers of G corresponding to polyconvex bulk energy densities of the form F([Xi] = 1/2[absolute value of Xi]┬▓ + h(det [Xi]), where the convex function h grows linearly at infinity."