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 $$-{\mathrm{\Delta }}_{\mathrm{\infty }}u-\beta \mathrm{\mid }Du\mathrm{\mid }=f,$$ subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.