An infinity Laplace equation with gradient term and mixed boundary conditions

We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation ―Δ ∞ u ― β|Du| = f, subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and super-solutions of a certain finite difference approximation.