Mike Crandall

A view of the theory of absolutely minimizing functions

A real-valued function f defined on an open subset U of $R^n$ is said to be absolutely minimizing if it is locally Lipschitz and has the following property: the Lipschitz constant of the restriction of f to any open subset V of U whose closure is a compact subset of U is not more than the Lipschitz constant of the restriction of f to the boundary of V. This notion was introduced by G. Aronsson in the 60's. Aronsson proved the existence of absolutely minimizing functions in U which continuously assumed given Lipschitz continuous boundary values on the boundary of U. He further demonstrated that if f is smooth, one could determine whether or not it was absolutely minimizing by checking if it solved a certain degenerate elliptic equation. This equation is nowadays called the ``infinity-Laplace'' equation. However, Aronsson also provided an example of an absolutely minimizing function which was not smooth. It wasn't until the 90's that the relation between the infinity-Laplace equation and absolutely minimizing functions was perfected by R. Jensen, who showed that the equation, understood in the viscosity sense, was indeed equivalent to the absolutely minimizing property and established uniqueness for the Dirichlet problem for the infinity-Laplace equation. It remains an open question if the regularity possessed by the example of Aronsson (Hölder continuous first derivatives) is enjoyed by all absolutely minimizing functions. We will introduce all of this and more, employing an elementary notion called ``comparison with cones'', which is equivalent to the absolutely minimizing property. Accessible open questions will be mentioned. The ``comparison with cones'' equivalence was noted by the speaker and L. C. Evans and R. Gariepy. This talk, which we hope will also be quite accessible, corresponds to an article in preparation with G. Aronsson and P. Juutinen.