LEGENDRE TRANSFORMS OF ELECTROSTATIC FREE-ENERGY FUNCTIONALS

In the Poisson-Boltzmann (PB) theory, the electrostatic free-energy functional of all possible electrostatic potentials for an ionic solution is often formulated in such a way that the Euler-Lagrange equation of such a functional is exactly the PB equation. However, such a PB functional is concave downward and maximized at its critical point, making it inconsistent in many applications where a macroscopic free-energy functional is minimized. Maggs [Europhys. Lett., 98 (2012), 16012] proposed a Legendre transformed form of the electrostatic free-energy functional of all possible dielectric displacements. This new functional is convex and minimized at the displacement corresponding to the critical point of the PB functional, and the minimum value is exactly the equilibrium electrostatic free energy. In this work, we study mathematically the Legendre transformed electrostatic free-energy functionals and the related variational principles. We first prove that the PB functional and its Legendre transformed functional are equivalent. We then consider a phenomeno-logical electrostatic free-energy functional that includes a higher-order gradient term, proposed by Bazant, Storey, and Kornyshev [Phys. Rev. Lett., 106 (2011), 046102] to describe charge-charge correlations. For such a functional, we introduce the corresponding Legendre transformed functional and obtain the related equivalence. We further consider the case without ions. We show that the electrostatic energy functional is equivalent to a Legendre transformed energy functional with constraint, and we prove the convergence of the Legendre transform of a perturbed electrostatic energy functional. Finally, we apply the Legendre transform to the dielectric boundary electrostatic free energy in molecular solvation.