Young measures, relaxation of functionals and existence results without weak lower semicontinuity (Q1977677)

The authors consider problems of the calculus of variations of the form \[ \inf\biggl\{\int_\Omega f(x,u(x),Du(x))dx\mid u-u_0\in W^{1,p}_0(\Omega;{\mathbb R}^m) \biggr\}, \] where \(\Omega\) is a bounded subset in \({\mathbb R}^N\), \(u\) is a mapping from \(\Omega\) to \({\mathbb R}^m\), \(Du\) is the derivative of \(u\), and \(u_0\) is a given function. The authors study the relaxation of the functional \(F(u)=\int_\Omega f(x,u(x),Du(x))dx\) by Young measures, when it is not lower semicontinuous for the weak topology of \(W^{1,p}(\Omega;{\mathbb R}^m)\). Optimality conditions are derived for the solutions to the corresponding relaxed problem. In particular, the case when \(N=1\) is carefully studied. Next, still for \(N=1\), using optimality conditions for the relaxed problems, sufficient conditions are given for which every solution of the relaxed problem is also a solution to the original one. This provides existence results for problems for which \(F\) is not necessarily weakly l.s.c. in \(W^{1,p}\). For related results see \textit{G. Aubert} and \textit{R. Tahraoui} [Differ. Integral Equ. 9, No. 1, 27-43 (1996; Zbl 0840.49001)], \textit{J.-P. Raymond} [J. Optimization Theory Appl. 82, No. 3, 571-592 (1994; Zbl 0808.49002)] and \textit{T. Roubiček} [``Relaxation in optimization and variational calculus'' (1997; Zbl 0880.49002)].