  This paper is a continuation of [13], where new variational principles were
introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue
here the program of using these Lagrangians to provide variational formulations
and resolutions to various basic equations and evolutions which do not normally
fit in the Euler-Lagrange framework. In particular, we consider stationary
equations of the form $ -Au\in \partial \phi (u)$ as well as i dissipative
evolutions of the form $-\dot{u}(t)-A_t u(t)+\omega u(t) \in \partial \phi (t,
u(t))$ were $\phi$ is a convex potential on an infinite dimensional space. In
this paper, the emphasis is on the cases where the differential operators
involved are not necessarily bounded, hence completing the results established
in [13] for bounded linear operators. Our main applications deal with various
nonlinear boundary value problems and parabolic initial value equations
governed by the transport operator with or without a diffusion term.
