A PDE Perspective of the Normalized Infinity Laplacian

The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [21] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [21] by the game theory. In this paper, the normalized infinity Laplacian, formally written as Delta(N)(infinity)u =broken vertical bar del u broken vertical bar(-2) Sigma(n)(i,j)=1 partial derivative(xi) u partial derivative(xj) u partial derivative(2) x(i)x(j)u, is defined in a canonical way with the second derivatives in the local maximum and minimum directions, and understood analytically by a dichotomy. A comparison with polar quadratic polynomials property, the counterpart of the comparison with cones property, is proved to characterize the viscosity solutions of the inhomogeneous normalized infinity Laplace equation. We also prove that there is exactly one viscosity solution of the boundary value problem for the infinity Laplace equation Delta(N)(infinity)u = f with positive f in a bounded open subset of R-n. The stability of the inhomogeneous infinity Laplace equation Delta(N)(infinity)u = f with strictly positive f and of the homogeneous equation Delta(N)(infinity)u = 0 by small perturbation of the right-hand-side and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [21].