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Author: Yanyan Li
Abstract: We study the regularity of the viscosity solution u of the σk-Loewner–Nirenberg problem on a bounded smooth domain Ω ⊂ Rn 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 n−2 of d 2 u are Ho ̈lder 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 n−2 derivatives and vanishes if and only if d 2 u is smooth in {0 ≤ d(x) < δ}. 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 ∂Ω contains more than one connected components, u is not differentiable in Ω. This is a joint work with Luc Nguyen and Jingang Xiong.
