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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.

An infinity Laplace equation with gradient term and mixed boundary conditions