Existence and uniqueness results for minimization problems with nonconvex functionals

Abstract

In some physical problems (mechanical problems, optimal control problems, phase transition problems, etc.), we have to minimize a functionalJ over a topological spaceU for whichJ is not sequentially lower semicontinuous. In this article, we prove new existence results for general one-dimensional vector problems of calculus of variations without any convexity condition on the integrand of the problem. In particular, we do not suppose that the integrand is split in two parts, one part depending on the gradient variable and the other part depending on the state variable, as is often supposed in recent results. In the case where the integrand is the sum of two functions, the first one depending on the gradient variable and the second one depending on the state variable, we also prove a uniqueness result without any convexity assumption with respect to the gradient variable.