Summary: We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation \(\partial_{x_i} u\partial_{x_j} u\partial^2_{x_i x_j} u= f\) in domains in \(\mathbb{R}^n\). We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition that \(f\) does not change its sign. We also discover a characteristic, property, which we call the comparison with standard functions property, of the viscosity sub- and 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 \(\partial_{x_i} u\partial_{x_j} u\partial^2_{x_i x_j}u= 0\), which states that 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.