Integral representation for functionals defined on $SBD^p$ in dimension two

We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(Omega)$, for $OmegasubsetR^2$ a bounded open Lipschitz set, $pin(1,infty)$. 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 $calH^{1}$-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.