Accepted Paper

Inserted: 15 oct 2017
Last Updated: 10 may 2019

Journal: Annali di Matematica Pura ed Applicata
Year: 2019

Abstract:

In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution, in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials $(\Psi_n)_n$ with superlinear growth at infinity and suitably converging to a $1$-positively homogeneous potential $\Psi$, and a smooth energy functional $\mathcal{E}$, we provide sufficient conditions on them ensuring that the solutions of the associated (generalized) gradient systems $(\Psi_n,\mathcal{E})$ converge as $n\to\infty$ to a Balanced Viscosity solution of the rate-independent system driven by $\Psi$ and $\mathcal{E}$. In specific cases, we also obtain results on the reverse approximation of Balanced Viscosity solutions by means of solutions to gradient systems. Our approach is based on the key observation that solutions to gradient systems and that Balanced Viscosity solutions to rate-independent systems can be characterized as (null-)minimizers of suitable trajectory functionals, for which we indeed prove Mosco-convergence. As particular cases, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems and their stochastic derivation.

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