On the convergence of integral functionals depending on vector-valued functions
The author considers the sequence of the integral functionals \[ (1)\quad F_ h(u,\Omega)=\int_{\Omega}f_ h(x,u(x),Du(x))dx,\quad u\in W^{1,p}_{loc}({\mathbb{R}}^ n,{\mathbb{R}}^ m), \] where the integrands \(f_ h:{\mathbb{R}}^ n\times {\mathbb{R}}^ m\times {\mathbb{R}}^{nm}\to {\mathbb{R}}\) are Carathéodory functions which satisfy a growth condition of the form \[ (2)\quad | \xi |^ p\leq f_ h(x,u,\xi)\leq s(1+| u|^ p+| \xi |^ p),\quad p\geq 1. \] A compactness and representation result for such a sequence is proved. Namely, under suitable assumptions (as for example quasiconvexity of integrands with respect to the last variable) there exists a subsequence which is \(\Gamma\)-convergent in the topology of \(L^ p(\Omega)\) to an integral functional of the same kind. Besides, it is proved that for any Lipschitz function u we have \(\Gamma (L^ p)\lim_{h}F_ h(u,\Omega)=\Gamma (L^{\infty})\lim_{h}F_ h(u,\Omega).\) A final example shows that without the coercivity assumption in (2) and without using \(\Gamma\)-convergence it is still possible to get similar results of convergence of integral functionals to a functional of the same type.
scientific article; zbMATH DE number 3897672