Regularity results for equilibria in a variational model for fracture

Abstract

SynopsisIn 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 formwhere Ω is an open, bounded subset of ℝN, q ≧1, g ∈ L∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\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 uΩ→ 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 corresponding to polyconvex bulk energy densities of the formwhere the convex function h grows linearly at infinity.