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Strong approximation of GSBV functions of piecewise smooth functions. (English) Zbl 0916.49002
Summary: Let \(\Omega\) be an open and bounded subset of \(\mathbb{R}^n\) with locally Lipschitz boundary. We prove that the functions \(v\in\text{SBV}(\Omega,\mathbb{R}^m)\) whose jump set \(S_v\) is essentially closed and polyhedral and which are of class \(W^{k,\infty}(\Omega\setminus \overline S_v,\mathbb{R}^m)\) for every integer \(k\) are strongly dense in GSBV\(^p(\Omega, \mathbb{R}^m)\), in the sense that every function \(u\) in GSBV\(^p(\Omega, \mathbb{R}^m)\) is approximated in \(L^p(\Omega,\mathbb{R}^m)\) by a sequence of functions \(\{v_j\}_{j\in\mathbb{N}}\) with the described regularity such that the approximate gradients \(\nabla v_j\) converge in \(L^p(\Omega, \mathbb{R}^{nm})\) to the approximate gradient \(\nabla u\) and the \((n-1)\)-dimensional measure of the jump sets \(S_{v_j}\) converges to the \((n-1)\)-dimensional measure of \(S_u\). The structure of \(S_v\) can be further improved in case \(p\leq 2\).

49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation
49M30 Other numerical methods in calculus of variations (MSC2010)
49N60 Regularity of solutions in optimal control
49Q20 Variational problems in a geometric measure-theoretic setting