  In this paper we discuss the effects of nontrivial boundary conditions or
backgrounds, including non-perturbative ones, on the renormalization program
for systems in two dimensions. Here we present an alternative renormalization
procedure such that these non-perturbative conditions can be taken into account
in a self-contained and, we believe, self-consistent manner. These conditions
have profound effects on the properties of the system, in particular all of its
$n$-point functions. To be concrete, we investigate these effects in the
$\lambda \phi^4$ model in two dimensions and show that the mass counterterms
turn out to be proportional to the Green's functions which have nontrivial
position dependence in these cases. We then compute the difference between the
mass counterterms in the presence and absence of these conditions. We find that
in the case of nontrivial boundary conditions this difference is minimum
between the boundaries and infinite on them. The minimum approaches zero when
the boundaries go to infinity. In the case of nontrivial backgrounds, we
consider the kink background and show that the difference is again small and
localized around the kink.
