On a Variational Inequality Containing a Memory Term with an Application in Electro-Chemical Machining

An obstacle problem with a memory term is studied in the framework of the variational inequality theory. Applying a fixed point argument and a convergence result for convex sets the existence and uniqueness of a solution are proved. Regularity results with respect to time and space are then deduced by using a penalization method. Furthermore, the time evolution of the solution is discussed. The particular evolutionary variational inequalities result from the application of a generalized Baiocchi-type transformation to degenerate free boundary problems with space- and, in particular, time-dependent coefficients. One of the applications is given by a quasi-stationary model for the electro-chemical machining problem.