10 papers
Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
Jiahuan Li, Shujun Shi
Let , , be a bounded convex domain, and let be the torsion function for the restricted half-Laplacian. We prove that is negative definit…
A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive -curvature Metrics
Jiahuan Li, Yilu Liu, Xi-Nan Ma
Let satisfy , and let $Σ^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\se…
The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity
Bin Deng, Jiahuan Li, Yilu Liu +1
Let and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{…
Global Minimality and Rigidity of the Constraint Map Vortex
Bin Deng, Jiahuan Li, Yilu Liu +1
We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figall…
A Local Lewy Theorem for -Harmonic Function with Non-zero Gradient in
Jiahuan Li, Yilu Liu, Xi-Nan Ma
We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space . Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(…
Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
Jiahuan Li, Yilu Liu, Xi-Nan Ma +1
Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such tha…
Strict Convexity for Solution of Liouville-Type Dirichlet Problems
Jiahuan Li, Shuning Xu
We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation , the real equation $σ…
Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional
Hongbin Cui, Jiahuan Li, Xiaowei Xu
We say that a hypersurface is -stationary if it is a critical point of the Euler-Dierkes-Huisken functional $\mathcal{E}_α(Σ)=\int_Σ|X|^α\, d\mathcal{…
Brunn--Minkowski Inequality for the First Complex -Hessian Eigenvalue
Chuanqiang Chen, Jiahuan Li, Xi-Nan Ma
There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the…
A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain
Jiahuan Li, Xi-Nan Ma, Paolo Salani
We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in , as an application we get t…