Nonconvex minimization related to quadratic double-well energy - approximation by convex problems

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied with taking into account the minimization of the corresponding integral functional. Such integral, as being not sequentially weakly lower semicontinuous, does not admit classical minimizers. To derive the relaxation formula for the infimum, the minimizing sequence consisting of solutions of convex problems appropriately approximating the original nonconvex one is constructed. The weak limit of this sequence together with the weak limit of the sequence of solutions of the corresponding dual problems and the weak limits of the characteristic functions related to the phase energies are involved in the relaxation formula.