arXiv subjects
Publications and source records attributed to YanYan Li.
Let $Ω$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $Ω$ such that for every $x$ in $G$ there is a unique closest point from $\partial Ω$ to $x$ (measured in the Finsler metric). We prove that the distance function from $\partial Ω$ is in $C^{k,α}_{loc}(G\cup \partial Ω)$, $k\ge 2$ and $0<α\le 1$, if $\partial Ω$ is in $C^{k,α}$.
This, and its sequel, concern some variations of a classical theorem of A.D. Alexandrov and teh Hopf Lemma.
We study solutions to conformally invariant equations with isolated singularties.
We establish Liouville type theorems for degenerate conformally invariant equations.
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.
We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.
We study the distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations.
We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
We establish a Liouville type theorem for some conformally invariant fully nonlinear equations
Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=\inf\|\nabla u\|_{L^2(\R^n)}\|u\|_{L^{2n/(n-2)}(\R^n)}^{-1}$ and A>0 is a constant depending on (M,g) only. The inequality is {\em sharp} in the sense that on any (M,g), $K$ can not be replaced by any smaller number and R_g can not be replaced by any continuous function which is smaller than R_g at some point. If (M,g) is not locally conformally flat, the exponent 2n/(n+2) can not be replaced by any smaller number. If (M,g) is locally conformally flat, a stronger inequality, with 2n/(n+2) replaced by 1, holds in all dimensions n>=3.