# A density result for GSBD and its application to the approximation of brittle fracture energies

created by paolini on 10 Dec 2012
modified by iurlano on 06 Jan 2015

[BibTeX]

Published Paper

Inserted: 10 dec 2012
Last Updated: 6 jan 2015

Journal: Calc. Var. Partial Differential Equation
Volume: 51
Pages: 315--342
Year: 2014
Doi: 10.1007/s00526-013-0676-7
We present an approximation result for functions $u:\Omega\to \mathbb{R}^n$ belonging to the space $GSBD(\Omega)\cap L^2(\Omega,\mathbb R^n)$ with $e(u)$ square integrable and $\mathcal{H}^{n-1}(J_u)$ finite. The approximating functions $u_k$ are piecewise continuous functions such that $u_k\to u$ in $L^2(\Omega,\mathbb{R}^n)$, $e(u_k)\to e(u)$ in $L^2(\Omega,\mathbb{M}^{n{\times}n}_{sym})$, $\mathcal{H}^{n-1}(J_{u_k}\triangle J_u)\to 0$, and $\int_{J_{u_k}\cup{J_u}}|u_k^\pm-u^\pm|\wedge1d\mathcal{H}^{n-1}\to0$. As an application, we provide the extension to the vector-valued case of the $\Gamma$-convergence result in $GSBV(\Omega)$ proved by Ambrosio and Tortorelli.