A variational principle for gradient flows

We verify – after appropriate modifications – an old conjecture of Brezis-Ekeland (3], 4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of self-duality inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value –which was the missing ingredient in 3]– and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.Both authors were partially supported by a grant from the Natural Science and Engineering Research Council of Canada.This paper is part of this authors Masters thesis under the supervision of the first named author.Revised version: 31 March 2004