The infinity Laplacian, Aronsson's equation and their generalizations

The infinity Laplace equation $\Delta_\infty u=0$ arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the $L^{\infty}$ variational problem of minimizing the functional ess-sup$_{U}\vert Du\vert.$ The more general functional ess-sup$_{U}F(x,u,Du)$ leads similarly to the so-called Aronsson equation $A_F[u]=0.$

In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to $L^{\infty}$ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional $L^{\infty}$ variational problems.