We prove the existence of variational solutions for
an evolution quasi-variational inequality with a first order
quasilinear operator and a variable convex set which is characterized
by a constraint on the absolute value of the gradient that depends
on the solution itself. The only required assumption on the nonlinearity
of this constraint is its continuity and positivity.
The method relies on an appropriate parabolic regularization and
suitable a priori estimates and allow us to obtain also the existence
of stationary solutions, by studying the asymptotic behaviour in time.
This is joint work with Lisa Santos.