Parabolic regularization of differential inclusions and the stop operator (Q1859330)

Parabolic differential inclusion with a set-valued term and a small ``diffusion'' coefficient \(\varepsilon\) at the elliptic operator is considered. The set-valued term is the subdifferential of the indicator function of a convex closed set in a finite-dimensional space. Along with the original inclusion the singular limit inclusion is considered, too. Under used assumptions the existence and the uniqueness of solutions for these inclusions are established. The strong convergence of solutions as \(\varepsilon\to 0\) to the solution of the singular limit inclusion is proved and the connection to elementary hysteresis operators is shown. The approach is based on a suitable penalty approximation. This problem arises for instance in multicomponent phase-field systems.