# zbMATH — the first resource for mathematics

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 $$p\leq 2$$.

##### MSC:
 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