Calculus of Variations and Geometric Measure Theory

S. Conti - M. Focardi - F. Iurlano

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

created by iurlano on 01 Oct 2015
modified by focardi on 31 Oct 2016


Archive for Rational Mechanics and Analysis

Inserted: 1 oct 2015
Last Updated: 31 oct 2016

Year: 2016


We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(\Omega)$, for $\Omega\subset \mathbb{R}^2$ a bounded open Lipschitz set, $p\in(1,\infty)$. 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 $\mathcal{H}^1$-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.