Global L∞L∞ gradient estimates for solutions to a certain degenerate elliptic equation

In view of the applications in the study of regularity properties of minimizers for a continuous model of transportation, which is a kind of divergence-constrained optimization problem, we prove a global L∞L∞ gradient estimate for solutions of an elliptic equation, whose ellipticity constants become degenerate at every point where |∇u|≤δ|∇u|≤δ, with δ>0δ>0. The exposition is as self-contained as possible.