Department of Mathematics, University of Stuttgart,
Allmandring 5b, 70569
Revised: 17 June 2014
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 ε and then taking the limit ε → 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 et al. [A. Mielke, R. Rossi and G. Savaré, Discrete Contin. Dyn. Syst. 2 (2010) 585–615; ESAIM: COCV 18 (2012) 36–80; To appear in J. Eur. Math. Soc. (2016)].
Mathematics Subject Classification: 49M99 / 49J20
Key words: Rate-independent systems / BV solutions / local minimizers / energy-dissipation balance
I would like to express my deep gratitude to Professor Giovanni Alberti for proposing the problem to me and giving many helpful discussions. I sincerely thank Professor Riccarda Rossi, Li-Chang Hung and Tran Minh-Binh for their helpful comments and remarks.
This work has been partially supported by the PRIN 2008 grant “Optimal Mass Transportation, Geometric and Functional Inequalities and Applications” and the FP7-REGPOT-2009-1 project “Archimedes Center for Modeling, Analysis and Computation”.
© EDP Sciences, SMAI 2016