Penalty method for a class of differential variational inequalities

The purpose of this paper is to investigate a class of differential variational inequalities involving a constraint set in Banach spaces. A well-posedness result for the inequality is obtained, including the existence, uniqueness, and stability of the solution in mild sense. Further, we introduce a penalized problem without constraints and prove that the solution to the original inequality can be approached, as a parameter converges to zero, by the solution of the penalized problem. Finally, an application to a comprehensive obstacle parabolic-elliptic system delineates the abstract results.