Integral Representation for Functionals Defined on SBDp in Dimension Two

We prove an integral representation result for functionals with growth conditions which give coercivity on the space , for , which is a bounded open Lipschitz set, and . The space SBD (p) of functions whose distributional strain is the sum of an L (p) part and a bounded measure supported on a set of finite -dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W (1,p) functions. We also obtain a generalization of Korn's inequality in the SBD (p) setting.