A new proof of semicontinuity by Young measures and an approximation theorem in Orlicz-Sobolev spaces (Q1773474)

The paper provides a weak lower semicontinuity result for a quasiconvex integral functional in the framework of Orlicz-Sobolev spaces. Let \(\Omega\) be a bounded open subset of \({\mathbb R}^n\), and let \(f:\Omega\times{\mathbb R}^m\times{\mathbb R}^{mn}\to[0,+\infty]\) be Carathéodory, such that for a.e. \(x\in\Omega\) and every \(s\in{\mathbb R}^m\), \(f(x,s,\cdot)\) is quasiconvex, and \[ f(x,s,z)\leq E(x,s)(1+\Phi(| z| )), \] for a Carathéodory \(E\) and an \(N\)-function \(\Phi\) satisfying additional qualitative assumptions. Then, the author proves that the integral \[ I(u)=\int_\Omega f(x,u(x),Du(x))dx \] is sequentially lower semicontinuous on \[ W^{1,\Phi,1}(\Omega,{\mathbb R}^m)=\left\{u\in W^{1,1}(\Omega,{\mathbb R}^m) : \int_\Omega \Phi(| Du(x)| )dx<+\infty\right\} \] along the sequences \(\{u_j\}\) converging to \(u\) in \(L^1_{\text{loc}}(\Omega,{\mathbb R}^m)\) such that \(\liminf_j\int_\Omega \Phi(| Du_j(x)| )dx\) is finite. The proof is based on a Jensen-type inequality for Young measures in Orlicz-Sobolev spaces proved in the same paper, and on a suitable approximation result for functions in \(W^{1,\Phi,1}(\Omega,{\mathbb R}^m)\) by Lipschitz ones.