Infinity-harmonic functions in the plane: Regularity by injectivity

Abstract

It has been a long standing conjecture that the $\infty$-harmonic functions in the plane have a 1/3-Hölder continuous gradient. It \emph{is} known that solutions are $C^1$ and that the gradient is locally $\alpha$-Hölder, but $\alpha$ comes without any positive lower bound. Aronsson's solution $x^{4/3}-y^{4/3}$ shows that no better general regularity is possible. In the plane there is also a connection between the $\infty$-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-Hölder continuity. Of course, an injective gradient is by no means a \emph{necessary} condition, as seen by smooth solutions such as the planes and cones.