The infinity Laplacian, Aronsson’s equation and their generalizations

The infinity Laplace equation Δ ∞ u = 0 Δ _∞ u=0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L ∞ L∞ variational problem of minimizing the functional ess-sup U | D u | . ess-supU|Du|. The more general functional ess-sup U F ( x , u , D u ) ess-supUF(x,u,Du) leads similarly to the so-called Aronsson equation A F [ u ] = 0. 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 ∞ L∞ 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 ∞ L∞ variational problems.