BV solutions constructed by epsilon-neighborhood method

TL;DR

This paper introduces a novel method for constructing BV solutions to rate-independent systems by employing epsilon-neighborhood local minimality and analyzing the limit as epsilon approaches zero, aligning with recent approaches by Mielke, Rossi, and Savaré.

Contribution

It presents a new epsilon-neighborhood method for constructing BV solutions, providing an alternative to vanishing viscosity techniques in rate-independent systems.

Findings

Solutions satisfy weak local stability.

Solutions obey a new energy-dissipation balance.

Method aligns with recent BV solution frameworks.

Abstract

We study a certain class of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order $\varepsilon$ and then taking the limit $\varepsilon \to 0$. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke, Rossi and Savar\'e.