Mathematics Colloquium: Minimizers for Nonconvex Variational Problems in the Plane via Convex/Concave Rearrangements

Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma.

In this paper we point out that such an approach was considered in the late 1930s in a series of papers by E. J. McShane for problems satisfying the usual coercivity condition.  Our goal is to survey some of McShane's results and compare them with Greco's work.  In addition, we will update some hypotheses that McShane made by making use of a result due to T. S. Angell on the avoidance of the Lavrentiev phenomenon. 

Free