Nonconvexity and Relaxation

Abstract

We have seen in the previous chapter that the weak lower semicontinuity property is crucial in order to employ the direct method of the calculus of variations to find minimizers of variational principles. This property is inherited by functionals whose integrands enjoy the appropriate convexity. Nonetheless, for an ever increasing number of interesting problems these convexity properties fail. In some cases, specific techniques may provide solutions to problems. In some others, this lack of convexity is a precursor of nonsolvability, at least in a classical sense. In the latter, highly oscillatory phenomena are usually involved. Parametrized measures were originally introduced by Young to account for oscillations in nonconvex optimal control problems where one could not reasonably expect classical solutions.