Gradient flows of non convex functionals in Hilbert spaces and applications

Issue		ESAIM: COCV Volume 12, Number 3, July 2006


Page(s)		564 - 614
DOI		10.1051/cocv:2006013
Published online		20 June 2006

ESAIM: COCV, July 2006, Vol. 12, pp. 564-614
DOI: 10.1051/cocv:2006013

Gradient flows of non convex functionals in Hilbert spaces and applications

Riccarda Rossi and Giuseppe Savaré

Dipartimento di Matematica "F. Casorati", Università di Pavia. Via Ferrata, 1 - 27100 Pavia, Italy; riccarda.rossi@unipv.it; giuseppe.savare@unipv.it

(Received May 3, 2005. / Published online: 20 June 2006)

Abstract
This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathcal{H}$

$\begin{displaymath}\begin{cases} u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t) &\text{{\it a.e.}\ in }(0,T), u(0)=u_0, \end{cases}\end{displaymath}$

where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures.

Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.

Mathematics Subject Classification. 35A15, 35K50, 35K85, 58D25, 80A22

Key words: Evolution problems, gradient flows, minimizing movements, Young measures, phase transitions, quasistationary models.

© EDP Sciences, SMAI 2006

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