  In this paper, we consider non-diffusive variational problems with mixed
boundary conditions and (distributional and weak) gradient constraints. The
upper bound in the constraint is either a function or a Borel measure, leading
to the state space being a Sobolev one or the space of functions of bounded
variation. We address existence and uniqueness of the model under low
regularity assumptions, and rigorously identify its Fenchel pre-dual problem.
The latter in some cases is posed on a non-standard space of Borel measures
with square integrable divergences. We also establish existence and uniqueness
of solutions to this pre-dual problem under some assumptions. We conclude the
paper by introducing a mixed finite-element method to solve the primal-dual
system. The numerical examples confirm our theoretical findings.
