Inhomogeneous infinity Laplace equation ✩

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂ 2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity suband supersolutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation ∂xi u∂xj u∂ 2 xixj u = 0, which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously. © 2007 Elsevier Inc. All rights reserved.