  This paper studies a {\it reversible} investment problem where a social
planner aims to control its capacity production in order to fit optimally the
random demand of a good. Our model allows for general diffusion dynamics on the
demand as well as general cost functional. The resulting optimization problem
leads to a degenerate two-dimensional bounded variation singular stochastic
control problem, for which explicit solution is not available in general and
the standard verification approach can not be applied a priori. We use a direct
viscosity solutions approach for deriving some features of the optimal free
boundary function, and for displaying the structure of the solution. In the
quadratic cost case, we are able to prove a smooth-fit $C^2$ property, which
gives rise to a full characterization of the optimal boundaries and value
function.
