A UNIQUENESS RESULT FOR A TWO DIMENSIONAL VARIATIONAL PROBLEM

We investigate the uniqueness of the solutions for a non-strictly convex problem in the Calculus of Variations of the form φ(∇v) − λv. Here, φ : R 2 → R is a convex function and λ is Lipschitz continuous. We prove the uniqueness when ∇λ is small and give some counterexamples when that is not the case. The proof is based on the global Lipschitz regularity of the minmizers and on the study of their level sets