Conditions for the integrand that are necessary and sufficient for the validity of a theorem on convergence with a functional (Q1363297)

The author indicates the conditions on the integral functional \[ F(u)= \int_\Omega L(x,u(x),\nabla u(x))dx \] in terms of the integrand \(L\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^q\), and \(u\in W^1_p(\Omega)\), \(p\geq 1\), that are necessary and sufficient for the relations \(F(u_n)\to F(u)\) and \(u_n\to u\) (weakly) in \(W^1_p(\Omega)\), in order to imply the strong convergence \(u_n\to u\) in \(W^1_p(\Omega)\). Similar results are obtained in the nongradient case \(F(u,\xi)= \int_\Omega L(x,u(x),\xi(x))dx\), where \(u\), \(\xi\in L_1(\Omega)\).