Calculus of Variations and Geometric Measure Theory

F. Iurlano

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


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
Links: online publication


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.

Keywords: Brittle fracture, free discontinuity problems, Generalized functions of bounded deformation