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Journal of Convex Analysis 23 (2016), No. 1, 053--075
Copyright Heldermann Verlag 2016

A Variational Principle for Gradient Flows of Nonconvex Energies

Goro Akagi
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

Ulisse Stefanelli
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
and: Istituto di Matematica Applicata, Via Ferrata 1, 27100 Pavia, Italy

We present a variational approach to gradient flows of energies of the form E = φ1 - φ2 where φ1, φ2 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.

Keywords: Evolution equations, gradient flow, nonconvex energy, variational formulation.

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