  In this paper, we study a new variant of Moreau's sweeping process with
velocity constraint. Based on an adapted version of Moreau's catching-up
algorithm, we show the well-posedness (in the sense existence and uniqueness)
of this problem in a general framework. We show the equivalence between this
implicit sweeping process and a quasistatic evolution variational inequality.
It is well known that the variational formulations of many mechanical problems
with unilateral contact and friction lead to an evolution variational
inequality. As an application, we reformulate the quasistatic antiplane
frictional contact problem for linear elastic materials with short memory as an
implicit sweeping process with velocity constraint. The link between the
implicit sweeping process and the quasistatic evolution variational inequality
is possible thanks to some standard tools from convex analysis and is new in
the literature.
