arXiv subjects

Haiyun Deng

Publications and source records attributed to Haiyun Deng.

9 recordsLinked to original sources

arXiv2026

Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites

In this paper, we determine the symmetry properties and non-vertex critical points of the second Neumann eigenfunction $u$, as well as the multiplicity of the corresponding eigenvalue, on isosceles trapezoids and kites. By exploiting reflection symmetry, we reduce the problem to a comparison between the second Neumann eigenvalue and the first mixed Dirichlet--Neumann eigenvalue on the half-domain. More precisely, for isosceles trapezoids with base angle $α\leq \fracπ{3}$, the second eigenfunction is antisymmetric. If $\fracπ{3}<α<\fracπ{2}$, there exists a critical height $\hat{h}(α)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(α)$ and symmetric when $h>\hat{h}(α)$, while at $h=\hat{h}(α)$ the second Neumann eigenvalue has multiplicity two. For a convex kite $P_1P_2P_3P_4$, where $P_1=(0,0)$, $P_2=(a,-h)$, $P_3=(1,0)$, and $P_4=(a,h)$, an analogous result holds: there exists a critical height $\tilde{h}(a)$ such that $u$ is symmetric with respect to the $x$-axis when $h<\tilde{h}(a)$ and antisymmetric with respect to the $x$-axis when $h>\tilde{h}(a)$, while at $h=\tilde{h}(a)$ the second Neumann eigenvalue has multiplicity two. In all cases where the second Neumann eigenvalue is simple, we determine all non-vertex critical points of the corresponding eigenfunction, thereby verifying the hot spots conjecture.

math.AP↗

arXiv2026

Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $μ_2(Ω)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $Ω$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the \emph{hot spots constant} $\mathfrak{C}(Ω)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(Ω)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(Ω) < 4$ for $Ω\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(Ω) < 11.2$ for $Ω\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure ``how wrong'' the \emph{hot spots conjecture} can be.

math.AP↗

arXiv2025

Stability and rigidity results of space-like hypersurface in the Minkowski space

In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs $x_{n+1}=u(x)$ over domain $Ω\subset\mathbb R^n$, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the boundary $\partial M$ lies on a hyperplane intersecting with constant angles, then the hypersurface must be a part of hyperboloid. Secondly, for convex space-like hypersurfaces with boundaries on a hyperboloid or light cone, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the angle function between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) on the boundary is constant, then such hypersurfaces must be a part of hyperboloid. These results significantly extend Gao's previous work presented in \cite{Gao1,Gao2}. Furthermore, we derive two fundamental integral identities for constant mean curvature (CMC) graphical hypersurfaces $x_{n+1}=u(x)$, $x\inΩ\subset\mathbb R^n$, and the boundary lies on a hyperplane. As some applications: we obtain complete equivalence conditions for hyperboloid identification through curvature properties. We also establish a geometric stability estimate demonstrating that the square norm of the trace-free second fundamental form $\bar h$ of $M$ is quantitatively controlled by geometric quantities of $\partialΩ$, as expressed by the inequality: $$ ||\bar h||_{L^2(Ω)}\leq C(n,K)||H_{\partialΩ}-H_0||_{L^1(\partialΩ)}^{1/2}. $$ Here, $H_{\partialΩ}$ is the mean curvature of $\partialΩ$, $H_0$ is some reference constant and $C$ is a constant. Finally, analogous estimates are established.

math.DG↗

arXiv2025

Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $Ω\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles.

math.AP↗

arXiv2024

On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains

In this paper, one of our aims is to investigate the instability of the distribution of the critical point set $\mathcal{C}(u)$ of a solution $u$ to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that $\mathcal{C}(u)$ in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set $\mathcal{C}(u)$ is unstable when any boundary of planar concentric circle annular domain $Ω$ has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets $u^{-1}_θ(0)(u_θ=\nabla u\cdot θ)$ and $u^{-1}(0)$, we prove that the solution $u$ on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that $\mathcal{C}(u)$ only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary $γ_E$ is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution $u$ by the moving plane method.

math.AP↗

arXiv2018

Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

In this paper, we mainly study the critical points and critical zero points of solutions $u$ to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distributions of connected components of the super-level sets $\{x\in Ω: u(x)>t\}$ and sub-level sets $\{x\in Ω: u(x)<t\}$ for some $t$, we obtain the geometric structure of interior critical point sets of $u$. Precisely, let $Ω$ be a multiply connected domain with the interior boundary $γ_I$ and the external boundary $γ_E$, where $u|_{γ_I}=ψ_1(x),~u|_{γ_E}=ψ_2(x)$. When $ψ_1(x)$ and $ψ_2(x)$ have $N_1$ and $N_2$ local maximal points on $γ_I$ and $γ_E$ respectively, we deduce that $\sum_{i = 1}^k {m_i}\leq N_1+ N_2$, where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$. In addition, when $\min_{γ_E}ψ_2(x)\geq \max_{γ_I}ψ_1(x)$ and $u$ has only $N_1$ and $N_2$ equal local maxima relative to $\overlineΩ$ on $γ_I$ and $γ_E$ respectively, we develop a new method to show that one of the following three results holds $\sum_{i = 1}^k {m_i}=N_1+N_2$ or $\sum_{i = 1}^k {m_i}+1=N_1+N_2$ or $\sum_{i = 1}^k {m_i}+2=N_1+N_2$. Moreover, we investigate the geometric structure of interior critical zero points of $u.$ We obtain that the sum of multiplicities of the interior critical zero points of $u$ is less than or equal to the half of the number of its isolated zero points on the boundaries.

math.AP↗

arXiv2018

Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $Ω$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane.

math.AP↗

arXiv2018

Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions

In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in Ω: u(x)>t\}$ for any $\mathop {\min}_{\partialΩ}u(x)<t<\mathop {\max}_{\partialΩ}u(x)$, we obtain the geometric structure of interior critical points of $u$. Precisely, when $Ω$ is simply connected, we develop a new method to prove $Σ_{i = 1}^k {m_i}+1=N$, where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$ and $N$ is the number of global maximal points of $u$ on $\partialΩ$. When $Ω$ is an annular domain with the interior boundary $γ_I$ and the external boundary $γ_E$, where $u|_{γ_I}=H,~u|_{γ_E}=ψ(x)$ and $ψ(x)$ has $N$ local (global) maximal points on $γ_E$. For the case $ψ(x)\geq H$ or $ψ(x)\leq H$ or $\mathop {\min}\limits_{γ_E}ψ(x)<H<\mathop {\max}\limits_{γ_E}ψ(x)$, we show that $Σ_{i = 1}^k {m_i} \le N$ (either $Σ_{i = 1}^k {m_i}=N$ or $Σ_{i = 1}^k {m_i}+1=N$).

math.AP↗

arXiv2017

Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of $\mathbb{R}^{n}(n\geq2)$. Secondly, we study the geometric structure about the critical set $K$ of solutions $u$ for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of $\mathbb{R}^{n}(n\geq3)$, and deduce that $K$ exists (respectively does not exist) a rotationally symmetric critical closed surface $S$. In fact, in an eccentric spherical annulus domain, $K$ is made up of finitely many isolated critical points ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis.

math.AP↗