A Variational Principle for Gradient Flows of Nonconvex Energies

We present a variational approach to gradient flows of energies of the form $E = \phi_1 - \phi_2$ where $\phi_1$, $\phi_2$ are convex functionals on a Hilbert space. A global parameter-dependent functional over trajectories is proved to admit minimizers. These minimizers converge up to subsequences to gradient-flow trajectories as the parameter tends to zero. These results apply in particular to the case of non λ-convex energies E. The application of the abstract theory to classes of nonlinear parabolic equations with nonmonotone nonlinearities is presented.