Variational methods play an essential role in many areas of mathematics such as analysis, geometry, or mathematical physics. One of the important goals is to understand properties of minimizers, or more generally critical points, of functionals represented by energy, entropy etc. Since the critical points are in often equivalent to solutions of differential equations, their analysis provide a valuable insight into dynamics of very complicated systems. Although basic properties include their existence, uniqueness, and regularity of critical points, the literature provide only very basic criteria for the uniqueness . To close this gap, we prove a unified and general criterion for the uniqueness of critical points of a functional in or without the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of
solutions to the corresponding Euler-Lagrange equations. To illustrate our method we present a unified proof of known results, as well as new theorems.

This is a joint work with Denis Bonheure, Ederson Moreira dos Santos, Alberto Saldana, and Hugo Tavares.