  Thin fluid or elastic films and membranes are found in nature and technology,
for instance, as confinements of living cells or in loudspeakers. When applying
a net force, resulting flows in an unbounded two-dimensional incompressible
low-Reynolds-number fluid or displacements in a two-dimensional linearly
elastic solid seem to diverge logarithmically with the distance from the force
center, which has led to some debate. Recently, we have demonstrated that such
divergences cancel when the total (net) force vanishes. Here, we illustrate
that, if a net force is present, the boundaries play a prominent role. Already
a single no-slip boundary regulates the flow and displacement fields and leads
to their decay to leading order inversely in distance from a force center and
the boundary. In other words, it is the boundary that stabilizes the system in
this situation, unlike the three-dimensional case, where an unbounded medium by
itself is able to absorb a net force. We quantify the mobility and
displaceability of an inclusion as a function of the distance from the
boundary, as well as interactions between different inclusions. In the case of
free-slip boundary conditions, a kinked boundary is necessary to achieve
stabilization.
