Calculus of Variations and Geometric Measure Theory

A. Mielke - R. Rossi - G. Savaré

Variational convergence of gradient flows and rate-independent evolutions in metric spaces

created by rossi on 11 Jul 2012


Accepted Paper

Inserted: 11 jul 2012
Last Updated: 11 jul 2012

Journal: Milan J. Math.
Year: 2012


We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials.

As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of $p$-gradient flows, $p>1$, when the exponents $p$ converge to $1$.