This dissertation is a study of second order, elliptic partial differential equations (PDE) that subject solutions to pointwise gradient constraints. These equations fall into the broad class of scalar non-linear PDE, and therefore, we interpret solutions in the viscosity sense and use methods from the theory of viscosity solutions. These equations are also naturally associated to free boundary problems as the boundary of the region where the gradient constraint is strictly satisfied cannot, in general, be determined before a solution of the PDE has been obtained. Consequently, we also employ techniques from PDE theory developed for free boundary problems.
In addition, we identify connections with control theory. Each solution of the PDE we consider has a probabilistic interpretation as an optimal value of a stochastic control problem. A distinguishing feature of these optimization problems is that the controlled processes have sample paths of bounded variation and thus may be ``singular" with respect to Lebesgue measure on the real line. The theory of stochastic singular control has been used to model spacecraft control, queueing systems, and financial markets in the presence of transaction costs. Our work makes considerable progress at rigorously interpreting the PDE that arise in these applications.