Relaxation of a variational problem

Summary

The relaxation of a variational problem replaces a functional $J$ that has no minimiser by its lower semicontinuous envelope $\bar J$. This is the largest lower semicontinuous functional below $J$. The relaxed problem has the same infimum as the original one. Every limit of a minimising sequence of the original problem, in the topology used, is a minimiser of the relaxed problem. For integral functionals, relaxation typically replaces the integrand by its convex or quasiconvex envelope in the gradient variable.