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Changyu Xia

Publications and source records attributed to Changyu Xia.

18 recordsLinked to original sources
arXiv2021

On Payne-Schaefer's Conjecture about an Overdetermined Boundary Problem of Sixth Order

This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of dimension no less than three. Secondly, we prove an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett's symmetry theorem. Finally, we prove a symmetry result for an overdetermined problem of second order by integral identities.

math.AP↗
arXiv2021

Isoperimetric Bounds for Lower Order Eigenvalues

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for further study are also proposed.

math.AP↗
arXiv2021

Estimates for eigenvalues of the Neumann and Steklov problems

We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

math.DG↗
arXiv2020

Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a Wentzell-type bi-Laplace problem on Euclidean balls.

math.AP↗
arXiv2020

On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-Pólya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.

math.AP↗
arXiv2019

Sharp Estimates for the First Eigenvalues of the Bi-drifting Laplacian

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.

math.DG↗
arXiv2018

Pohozaev identity for the anisotropic $p$-Laplacian and estimates of torsion function

In this paper we prove the Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of our identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb{R}^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded domains of $\mathbb{R}^n$, some sharp estimates for the torsion function of compact manifolds with boundary and a nonexistence result for the solutions of the Laplace equation on closed Riemannian manifolds.

math.AP↗
arXiv2017

The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold and Finsler space which support a Caffarelli-Kohn-Nirenberg inequality.

math.DG↗
arXiv2015

Eigenvalues of the Wentzell-Laplace Operator and of the Fourth Order Steklov Problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Stekolv problems on a Euclidean ball.

math.AP↗
arXiv2015

On the H-almost ricci soliton

We introduce the concept {\it $h$-almost Ricci soliton} which extends naturally the {\it almost Ricci soliton} by Pigola-Rigoli-Rimoldi-Setti and show that a compact nontrivial $h$-almost Ricci soliton of dimension no less than three with $h$ having defined signal and constant scalar curvature is isometric to a standard sphere with the potential function well determined. We also consider the {\it $h$-Ricci soliton} which is a particular case of the $h$-almost Ricci soliton and a generalization of the {\it Ricci soliton} and give characterizations for a special class of gradient $h$-Ricci solitons.

math.DG↗
arXiv2010

Inequalities for eigenvalues of the buckling problem of arbitrary order

This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimates [16] on the buckling eigenvalues of order two to arbitrary order.

math.DG↗
arXiv2010

Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &Δ{\mathbf u}+ α{\rm grad}(\text{div}{\mathbf u})=-σ{\mathbf u}, \ \text{in $Ω$}, &{\mathbf u}|_{\partial Ω}={\mathbf 0}. \aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\text{th}}$ eigenvalue $σ_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature.

math.DG↗
arXiv2010

Inequalities for Eigenvalues of the Buckling Problem of Higher Orders

This paper studies eigenvalues of the buckling problem of arbitrary order on compact domains in Euclidean spaces and spheres. We prove universal bounds for the $k$-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthens the recent work by Jost, Li-Jost, Wang and Xia and generalizes Cheng-Yang's recent estimates on the buckling eigenvalues of order two to arbitrary order.

math.AP↗
arXiv2010

Inequalities for the Steklov Eigenvalues

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $Ω$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=λ_0 < λ_1\leq λ_2\leq ... $ denote the eigenvalues of the Steklov problem: $Δu=0$ in $Ω$ and $(\partial u)/(\partial ν)=λu$ on $\partial Ω$. Then $\sum_{i=1}^{n} λ^{-1}_i \geq (n^2|Ω|)/(|\partialΩ|) $ with equality holding if and only if $Ω$ is isometric to an $n$-dimensional Euclidean ball. Let $M$ be an $n(\geq 2)$-dimensional compact connected Riemannian manifold with boundary and non-negative Ricci curvature. Assume that the mean curvature of $\pa M$ is bounded below by a positive constant $c$ and let $q_1$ be the first eigenvalue of the Steklov problem: $ Δ^2 u= 0$ in $ M$ and $u= (\partial^2 u)/(\partial ν^2) -q(\partial u)/(\partial ν) =0$ on $ \partial M$. Then $q_1\geq c$ with equality holding if and only if $M $ is isometric to a ball of radius $1/c$ in ${\bf R}^n$.

math.SP↗
arXiv2009

Universal Bounds for Eigenvalues of the Polyharmonic Operators

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-Pólya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a Euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by Ashbaugh. Finally, we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere.

math.DG↗