Lower semicontinuity of quasi-convex integrals in BV
The paper deals with the lower semicontinuity of integral functionals \(F(u)=\int_\Omega f(\nabla u) dx\) (with \(\Omega\subset{\mathbb R}^m\) open, \(u:\Omega\to{\mathbb R}^n\)) under the assumption that \(f\) is quasi convex and satisfies a linear growth condition from below and a \(q\)-growth condition from above, namely \[ c(| p| -1)\leq f(p)\leq C(| p| ^q+1)\qquad\qquad\forall p\in M^{n\times m}. \] The main result of the paper, obtained under the assumption that \(q<m/(m-1)\), is that \[ \liminf_{h\to\infty}\int_\Omega f(\nabla u_h) dx\geq\int_\Omega f(\nabla u) dx\tag{*} \] whenever \((u_h)\subset W^{1,1}(\Omega,{\mathbb R}^n)\) converges to \(u\) in \(L^1_{\text{loc}}\). In (*) \(\nabla u\) is understood as the approximate differential of the BV function \(u\). This result implies in particular an estimate from below of the relaxed functional of \(F\) in the space \(\text{BV}(\Omega,{\mathbb R}^n)\). The proof relies on the blow-up technique of I. Fonseca and S. Müller and on sharp estimates for suitable extension operators.
scientific article; zbMATH DE number 1221537