Relaxation in BV of integrals with superlinear growth (Q2926569)

Let \(\Omega\subset \mathbb{R}^{n}\) be a bounded, open set, where \(n\geq 2\). Consider the variational integral NEWLINE\[NEWLINE F(u, \Omega):=\int_{\Omega}f(\nabla u(x))dx, NEWLINE\]NEWLINE where \(u\in BV(\Omega;\mathbb{R}^{N})\) and \(f:\mathbb{R}^{N\times n}\rightarrow \mathbb{R}\) is a quasiconvex function satisfying the growth condition NEWLINE\[NEWLINE 0\leq f(\xi)\leq L(1+|\xi|^{r}) NEWLINE\]NEWLINE for \(r\in [1,\frac{n}{n-1}).\)NEWLINENEWLINEIn this paper the following inequality is proved NEWLINE\[NEWLINE \Gamma_{loc}(u,\Omega)\geq F(u, \Omega) +\int_{\Omega} f^{\infty}\Big(\frac{D^{s}u}{|D^{s}u}|\Big) |D^{s}u|, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Gamma_{loc}(u,\Omega)=\inf_{(u_j)}\Big \{\lim\inf_{j\rightarrow \infty}F(u_j, \Omega): \;(u_j)\subset W^{1,r}_{loc}(\Omega,\mathbb{R}^{N}),\; u_j\rightharpoonup^{*}u NEWLINE\]NEWLINE NEWLINE\[NEWLINE \text{in}\; BV(\Omega,\mathbb{R}^{N})\Big \} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE f^{\infty}(\xi):=\overline{\lim_{t\rightarrow \infty}}\frac{f(t\xi)}{t}<\infty. NEWLINE\]