Minimal Lipschitz Extensions and the Finsler Infinity Laplacian

The minimal Lipschitz extension problem is the prototypical example of an L^{\infty} variational problem, the study of which was initiated by Aronsson in the 1960's. When the Lipschitz seminorm under consideration is defined relative to the Euclidean norm, a well-developed theory shows that so-called absolutely minimizing Lipschitz extensions are characterized by the infinity Laplace equation, an L^{\infty} version of the classical Laplace equation. In 2004, Aronsson, Crandall, and Juutinen observed that, when the Euclidean norm is replaced by an arbitrary (Finsler) norm, absolutely minimizing Lipschitz extensions solve a PDE involving a potentially very complicated, discontinuous elliptic operator, the Finsler infinity Laplace equation. Since then, proving absolute minimality of solutions of this and related PDE has been an important question in the L^{\infty} calculus of variations. In this talk, I discuss recent work establishing that the Finsler infinity Laplace equation does, in fact, characterize absolutely minimizing Lipschitz extensions.