Quasiconvexity and relaxation of nonconvex problems in the calculus of variations (Q798943)

The author considers the problem \(\inf\{f(\nabla u(x))dx;\quad u\in u_ 0+W_ 0^{1,p}(\Omega,R^ m)\},\) where \(\Omega\subset R^ n\) is a bounded domain and \(f:R^{nm}\to R\) is continuous. From the main result of the paper one can obtain that, under some additional coercivity conditions, the relaxed problem, in which f is replaced by its lower quasi-convex envelope, has optimal solutions and the values of the two problems are equal.