Abstract

We present a variational approach to gradient flows of energies of the form E=ϕ1−ϕ2E = \phi_1 - \phi_2 where ϕ1\phi_1, ϕ2\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.

Author information

Contact details are reproduced from the original publication and may be historical.

Goro Akagi

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

akagi@port.kobe-u.ac.jp

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
and: CNR

ulisse.stefanelli@univie.ac.at

Keywords

  • Evolution equations
  • gradient flow
  • nonconvex energy
  • variational formulation

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Suggested citation

G. Akagi, U. Stefanelli. “A Variational Principle for Gradient Flows of Nonconvex Energies.” Journal of Convex Analysis 23 (2016), No. 1, 53–75. https://doi.org/10.68381/jca23003

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