8 problems

1/3-Hölder gradient conjecture. Infinity-harmonic functions in the plane should have a $1/3$-Hölder continuous gradient.

Let $u_p$ be $p$-harmonic functions and let $u_\infty$ be the corresponding infinity-harmonic limit, with the functions considered in a setting where their gradients are positive.…

Infinity-harmonic regularity conjecture. Infinity-harmonic functions have $C^{1,1/3}$ regularity and also belong to $W^{2,q}$ for every $q<3/2$.

Let $M$ be the manifold under consideration, and let $u:M\to S^1$ be an $\infty$-harmonic map in a fixed homotopy class. Uniqueness conjecture. The map $u$ is unique up to rotation…

Let $(M,g)$ be a Riemannian manifold, and consider $\infty$-harmonic functions and the associated notions of viscosity solutions, comparison with cones, gradient estimates, and reg…

Conjecture. The gradient magnitude satisfies

Streamline and Cl-point conjecture. Every streamline except the medians joins a diagonal before reaching the midpoint, and the only Cl-points are the points on the diagonals.

Let $u$ be a planar $\infty$-harmonic function. The preceding results establish Sobolev regularity for powers of $|Du|$ and measure bounds for the distributional determinant, motiv…