Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem

Abstract

We study the regularity of the viscosity solution u of the σ k -Loewner-Nirenberg problem on a bounded smooth domain Ω ⊂ R n for k ≥ 2. It was known that u is locally Lipschitz in Ω. We prove that, with d being the distance function to ∂Ω and δ > 0 sufficiently small, u is smooth in {0 < d(x) < δ} and the first (n−1) derivatives of d n−2 2 u are Hölder continuous in {0 ≤ d(x) < δ}. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of ∂Ω and its covariant derivatives and vanishes if and only ifUsing 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 ∂Ω contains more than one connected components, u is not differentiable in Ω.

Interior Gradient Estimates for General Prescribed Curvature Equations

“…It can be used to establish existence results for Dirichlet problems with less regular boundary data or seek locally Lipschitz continuous viscosity solutions on noncompact domains. For the significance of C 1 estimates (versus C 2 estimates), one can see from the σ k Loewner-Nirenberg problem that, for 2 ≤ k ≤ n, there are nonexistence results of C 2 solutions (see [12,13]), so no C 2 solutions is available (if C 2 estimates hold, higher derivative estimates hold, and one would obtain smooth solution). The existence of Lipschitz continous viscosity solutions to the σ k Loewner-Nirenberg problem relies on C 1 estimates (see [6]).…”