A counterexample of a uniqueness result

Abstract

Uniqueness of minimizers of constrained or unconstrained energy functionals is a very subtle issue that has attracted many mathematicians in the last decades. This interest is motivated by several reasons: The uniqueness implies that the critical point inherits all the symmetry and monotonicity properties of the problem. For example if the functional and its domain are radially decreasing, then is the critical point. This considerably reduces the difficulty of the study of quantitative properties of the underlying PDE, which is reduced to an ODE. Uniqueness also “guarantees” the stability, and simplifies the dynamics of the gradient flow induced by the functional.There are very few general results dealing with uniqueness of critical points in the literature. The purpose of this paper is to provide a counterexample of a general result obtained by B. Dacorogna in his book “Introduction to the calculus of variations, Edition 2004”.