Energy minimization is the process by which nature selects the configuration of a system and balances the relevant forces. But not all of the current standard energy functionals in electrostatics are bounded below, leading to a contradiction. By recognizing Legendre transforms at the stationary points (or maximums) of non-convex (or concave) functionals, one can rewrite the functionals in terms of the transformed variable such that the new functionals are convex, hence better suited to standard optimization techniques. In this talk I will describe how Legendre transforms can be applied to reformulate the electrostatic free energy functional and present a proof of the equivalence of the new formulation with the classical one. I will also discuss how the Legendre transformed functional can be applied to the dielectric boundary problem in molecular solvation.