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
