An approximation result for special functions with bounded deformation (Q1887198)

A \textit{special displacement with bounded deformation} is a function \(u:\Omega \subset \mathbb R^N \to \mathbb R^N\) whose symmetrized gradient is a bounded measure which coincides, outside an \((N-1)\)-dimensional rectifiable \textit{jump set} \(J_u\), with a summable function \(e(u)\). In the paper the author proves that in dimension \(N=2\) if \(u\) and \(e(u)\) are square integrable and \({\mathcal H}^{N-1}(J_u)\) is finite then \(u\) can be approximated with a sequence \(\{u_n\}\) of piecewise continuous functions whose jump sets \(J_{u_n}\) are relatively closed, with \(u_n\) and \(e(u_n)\) converging strongly in \(L^2\), respectively, to \(u\) and \(e(u)\) and the lengths \({\mathcal H}^{N-1}(J_{u_n})\) converging to \({\mathcal H}^{N-1}(J_u)\). As an application it is given an Ambrosio-Tortorelli type approximation of a functional which appears in the theory of brittle fracture in linearized elasticity. In the addendum the results are extended to dimension \(N\geq 3\) by a careful application of a discretization technique.