  In this paper we consider a free boundary problem in the 3-dimensional
Lorentz-Minkowski space $\l^3$ which deals spacelike surfaces whose mean
curvature is a linear function of the time coordinate and the boundary moves in
a given support plane. We study spacelike surfaces that project one-to-one into
a strip of the support and that locally are critical points of a certain energy
functional involving the area of the surface, a timelike potential and
preserves the volume enclosed by the surface. We call these surfaces stationary
bands. We establish existence of such surfaces and we investigate their
qualitative properties. Finally, we give estimates of its size in terms of the
initial data.
