As part of the MSU Mathematics Lecture Series, Professor Dean Carlson, from the American Mathematical Society, will be presenting : “Minimizers for Nonconvex Variational Problems in the Plane via Convex/Concave Rearrangements”. This presentation is scheduled for Wednesday November 2nd at 3:30PM in Cheek 301. Everyone is welcome to attend!
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 presentation, we point out that such an approach was considered in the late 1930’s 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.