Legendre transform is a basic notion for representing different quantities in physics. It is also a useful method in optimization. Recently, Maggs [Europhys. Lett. 98, 16012, 2012] proposed a Legendre transformed electrostatic free-energy functional. After an introduction of the Legendre transform, in this talk, I will present a proof of the equivalence of the Legendre transformed functional and the original one. I will also describe some of the extensions, in particular to a variational theory of molecular solvation. This is joint work with Benjamin Ciotti.

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