Weak solutions to rate-independent systems: Existence and regularity

Weak solutions for rate-independent systems has been considered by many authors recently. In this thesis, I shall give a careful explanation (benefits and drawback) of energetic solutions (proposed by Mielke and Theil in 1999) and BV solutions constructed by vanishing viscosity (proposed by Mielke, Rossi and Savare in 2012). In the case of convex energy functional, then classical results show that energetic solutions is unique and Lipschitz continuous. However, in the case energy functional is not convex, there is very few results about regularity of energetic solutions. In this thesis, I prove the SBV and piecewise C^1 regularity for energetic solution without requiring the convexity of energy functional. Another topic of this thesis is about another construction of BV solutions via epsilon-neighborhood method.