Regularity of viscosity solutions of the ${\sigma }_k$-Loewner-Nirenberg problem

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 $k\ge 2$. 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 Hölder continuous in $\{0\le d(x)<\delta \}$. 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 $\{0\le d(x)<\delta \}$. 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 $\Omega$.

Cited by in corpus (1)

• The ${\sigma }_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$