Minimal Lipschitz Extensions and the Finsler Infinity Laplacian

  • Peter Morfe (MPI MiS, Leipzig)
E1 05 (Leibniz-Saal)


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.

Katja Heid

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Felix Pogorzelski

Universität Leipzig