Gamma-convergence of gradient flows on Hilbert and metric spaces and applications

Pages: 1427 - 1451, Volume 31, Issue 4, December 2011

doi:10.3934/dcds.2011.31.1427       Abstract        References        Full Text (496.4K)       Related Articles

Sylvia Serfaty - UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France (email)

Abstract: We are concerned with $\Gamma$-convergence of gradient flows, which is a notion meant to ensure that if a family of energy functionals depending of a parameter $\Gamma$-converges, then the solutions to the associated gradient flows converge as well. In this paper we present both a review of the abstract "theory" and of the applications it has had, and a generalization of the scheme to metric spaces which has not appeared elsewhere. We also mention open problems and perspectives.

Keywords:  Gradient flow, Gamma-convergence, Ginzburg-Landau equation, Allen-Cahn equation, Cahn-Hilliard equation, Mullins-Sekerka evolution.
Mathematics Subject Classification:  Primary: 35K15, 35K99; Secondary: 35Q57, 35Q56.

Received: May 2009;      Revised: July 2010;      Published: September 2011.

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