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Guangyue Huang

Publications and source records attributed to Guangyue Huang.

30 records · Page 2Linked to original sources
arXiv2015

Gradient estimates and Liouville type theorems for a nonlinear elliptic equation

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$Δu+au\log u=0,$$ where $a$ is a nonzero constant. In particular, for $a<0$, we prove that any bounded positive solution of the above equation with a suitable condition for $a$ with respect to the lower bound of Ricci curvature must be $u\equiv 1$. This generalizes a classical result of Yau.

math.DG↗
arXiv2015

Estimates for eigenvalues of the operator $L_r$

In this paper, we consider an eigenvalue problem of the elliptic operator $$ L_r={\rm div}(T^r\nabla\cdot )$$ on compact submanifolds in arbitrary codimension of space forms $\mathbb{R}^N(c)$ with $c\geq0$. Our estimates on eigenvalues are sharp.

math.DG↗
arXiv2012

Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=Δ_ϕ(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvature

math.DG↗
arXiv2011

On a classification of the quasi Yamabe gradient solitons

In this paper, we introduce the concept of quasi Yamabe gradient solitons, which generalizes the concept of Yamabe gradient solitons. By using some ideas in [7,8], we prove that $n$-dimensional $(n\geq3)$ complete quasi Yamabe gradient solitons with vanishing Weyl curvature tensor and positive sectional curvature must be rotationally symmetric. We also prove that any compact quasi Yamabe gradient solitons are of constant scalar curvature.

math.DG↗
arXiv2011

Gradient estimates for the porous medium equations on Riemannian manifolds

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=Δu^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez and Villani in [10]. Moreover, our results recover the ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].

math.DG↗
arXiv2009

Estimates for the higher order buckling eigenvalues in the unit sphere

We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere $(-Δ)^p u=Λ(-Δ) u$ with order $p(\geq2)$. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalues independent of the domains. In particular, for $p=2$, our result is sharp than estimates on eigenvalues of the buckling problem obtained by Wang and Xia.

math.DG↗
arXiv2009

Estimates for lower order eigenvalues of a clamped plate problem

For a bounded domain $Ω$ in a complete Riemannian manifold $M^n$, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.

math.DG↗