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Gradient flows of non convex functionals in Hilbert spaces and applications

Published online by Cambridge University Press:  20 June 2006

Riccarda Rossi  and
Giuseppe Savaré
Riccarda Rossi
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1 – 27100 Pavia, Italy; riccarda.rossi@unipv.it; giuseppe.savare@unipv.it
Giuseppe Savaré
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1 – 27100 Pavia, Italy; riccarda.rossi@unipv.it; giuseppe.savare@unipv.it
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Abstract

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] 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 theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.

Keywords

Evolution problemsgradient flowsminimizing movementsYoung measuresphasetransitionsquasistationary models.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 564 - 614
Copyright
© EDP Sciences, SMAI, 2006

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