Regularity of viscosity solutions of the $\sigma_k$-Loewner-Nirenberg problem

TL;DR

This paper investigates the regularity of viscosity solutions to the $\sigma_k$-Loewner-Nirenberg problem, establishing conditions for smoothness near the boundary and identifying invariants related to boundary geometry.

Contribution

It proves the smoothness of solutions near the boundary under certain conditions and introduces a boundary invariant involving principal curvatures and their derivatives.

Findings

Solutions are smooth near the boundary when a boundary invariant vanishes.

The boundary invariant is a polynomial of principal curvatures and their derivatives.

Solutions are not differentiable in the interior if the boundary has multiple components.

Abstract

We study the regularity of the viscosity solution $u$ of the $\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $\Omega$. We prove that, with $d$ being the distance function to $\partial\Omega$ and $\delta > 0$ sufficiently small, $u$ is smooth in $\{0 < d(x) < \delta\}$ and the first $(n-1)$ derivatives of $d^{\frac{n-2}{2}} u$ are H\"older continuous in $\{0 \leq d(x) < \delta\}$. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of $\partial\Omega$ and its covariant derivatives and vanishes if and only if $d^{\frac{n-2}{2}} u$ is smooth in $\{0 \leq d(x) < \delta\}$. Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when $\partial\Omega$ contains more than one connected components, $u$ is not differentiable in $\Omega$.