arXiv subjects

Ruofei Yao

Publications and source records attributed to Ruofei Yao.

8 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 $\alpha\leq \frac{\pi}{3}$, the second eigenfunction is antisymmetric. If $\frac{\pi}{3}<\alpha<\frac{\pi}{2}$, there exists a critical height $\hat{h}(\alpha)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(\alpha)$ and symmetric when $h>\hat{h}(\alpha)$, while at $h=\hat{h}(\alpha)$ 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.

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arXiv2026

Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter

We study the first nontrivial Steklov eigenvalue of perimeter-normalized regular $N$-gons and show that it is strictly increasing in $N$. The proof mainly relies on an analytic framework that establishes a refined asymptotic expansion in three steps: first, identifying the Steklov eigenvalue as the maximal eigenvalue of a Toeplitz-type operator; second, deriving the eigenvalue and its associated eigenfunctions simultaneously via Schur reduction; and finally, obtaining the exact coefficients in the Schur moment expansion by evaluating Euler-type sums. The monotonicity is proved to be eventual, holding for $N\ge 20$. For the remaining cases $3\le N\le 20$, we provide complementary computer-assisted verification, confirming monotonicity across the full range of $N$.

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arXiv2026

Symmetry properties for positive solutions of mixed boundary value problems in a sub-spherical sector

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or narrow width, thereby enabling the application of the moving plane method. Secondly, in contrast to the purely Dirichlet case, a key challenge is to establish the non-vanishing of the tangential derivative of $u$ along the Neumann boundary. To address this, we employ local analysis techniques of angular derivatives, as introduced by Hartman and Wintner [Amer. J. Math., 1953]. Thirdly, we identify the signs of directional derivatives of $u$ along sections of the moving line. Using a planar sub-spherical sector as an example, we illustrate how these new innovative techniques and the moving plane method can be combined to derive symmetry and monotonicity results, particularly when the amplitude is less than or equal to $2\pi/3$.

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arXiv2025

Monotonicity properties of the Robin torsion function in a class of symmetric planar domains

We prove the monotonicity property of the Robin torsion function in a smooth planar domain $\Omega$ with a line of symmetry, provided that the Robin coefficient $\beta$ is greater than or equal to the negative of the boundary curvature $\kappa$ (i.e., $\beta \geq -\kappa$ on $\partial\Omega$). We also show that this condition is, in a certain sense, sharp by constructing a counterexample.

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arXiv2024

On the location of the maximal gradient of the torsion function over some non-symmetric planar domains

We investigate the location of the maximal gradient of the torsion function on certain non-symmetric planar domains. First, by establishing uniform estimates for convex narrow domains, we show that as a planar domain bounded by two graphs becomes increasingly narrow, the location of the maximal gradient of its torsion function converges to the endpoints of the longest vertical segment, with smaller curvature among them. This result confirms that Saint-Venant's conjecture on the location of fail points holds for asymptotically narrow domains. Second, for triangles, we prove that the maximal gradient of the torsion function always occurs on the longest side, lying between the foot of the altitude and the midpoint of that side. Moreover, via nodal line analysis, we show that, restricted to each side, the critical point of the gradient is unique and non-degenerate. Additionally, by perturbation and barrier arguments, we establish that for a class of nearly equilateral triangles, this critical point is closer to the midpoint than to the foot of the altitude, and the maximal gradient at the midpoint exceeds that at the foot of the altitude. Third, employing the reflection method, we demonstrate that for a non-concentric annulus, the maximal gradient of the torsion function is always attained at the point on the inner boundary closest to the center of the outer boundary.

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arXiv2024

Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles

We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on the remaining two sides. Using the method of moving planes, we prove that if the Neumann vertex is non-obtuse, then every positive solution is strictly increasing in the direction of the inward unit normal to the Dirichlet side. If the Neumann vertex is obtuse, we show that monotonicity instead holds in the direction of the outward normal direction to the longer Neumann side, under certain technical conditions. Furthermore, by applying the maximum principle, we demonstrate that these monotonicity properties extend to the first mixed eigenfunction of the Laplacian: its unique global maximum lies on the longer Neumann side and coincides with the Neumann vertex precisely when that vertex is non-obtuse or when the Neumann sides have equal length. This answers a question raised in the Polymath7 research thread 1 regarding the location of extrema for mixed boundary eigenfunctions in triangles.

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arXiv2023

Uniqueness of critical points of the second Neumann eigenfunctions on triangles

This paper investigates the second Neumann eigenfunction $u$ of a planar triangle $T$. In a recent paper by Judge and Mondal [Ann. Math., 2022], it was shown that $u$ has no critical points in the interior of $T$. In this paper, we show that $u$ has at most one non-vertex critical point and that $u$ is monotone in a certain direction in $T$. More precisely, when $T$ is not equilateral, we show that $u$ vanishes at some vertex if and only if $T$ is superequilateral, and that $u$ has a non-vertex critical point if and only if $T$ is acute and not superequilateral. These results confirm both the original theorem and Conjecture 13.6 of Judge and Mondal [Ann. Math., 2020]. We also resolve the objective of Polymath 7 (research thread 1), namely, that the extrema of $u$ are attained only at the endpoints of the longest side. In addition, we settle a conjecture of Siudeja [Proc. Amer. Math. Soc., 2016] on the ordering of mixed Dirichlet--Neumann Laplacian eigenvalues for triangles. Our proofs combine the continuity method, eigenvalue inequalities, the maximum principle, and the moving plane method.

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arXiv2022

On Laplacian eigenvalue equation with constant Neumann boundary data

Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study boundary behaviors of solutions to the Laplacian eigenvalue equation with constant Neumann data. \begin{align} \label{cequation0} \begin{cases} -\Delta u=cu\quad &\mbox{in $\Omega$}\\ \frac{\partial u}{\partial \nu}=-1\quad &\mbox{on $\partial \Omega$}. \end{cases} \end{align}First, by using properties of Bessel functions and proving new inequalities on elementary symmetric polynomials, we obtain the following inequality for rectangular boxes, balls and equilateral triangles: \begin{align} \label{bbb} \lim_{c\rightarrow \mu_2^-}c\int_{\partial \Omega}u_c\, d\sigma\ge \frac{n-1}{n}\frac{P^2(\Omega)}{|\Omega|}, \end{align}with equality achieved only at cubes and balls. In the above, $u_c$ is the solution to the eigenvalue equation and $\mu_2$ is the second Neumann Laplacian eigenvalue. Second, let $\kappa_1$ be the best constant for the Poincar\'e inequality with mean zero on $\partial \Omega$, and we prove that $\kappa_1\le \mu_2$, with equality holds if and only if $\int_{\partial \Omega}u_c\, d\sigma>0$ for any $c\in (0,\mu_2)$. As a consequence, $\kappa_1=\mu_2$ on balls, rectangular boxes and equilateral triangles, and balls maximize $\kappa_1$ over all Lipschitz domains with fixed volume. As an application, we extend the symmetry breaking results from ball domains obtained in Bucur-Buttazzo-Nitsch[J. Math. Pures Appl., 2017], to wider class of domains, and give quantitative estimates for the precise breaking threshold at balls and rectangular boxes. It is a direct consequence that for domains with $\kappa_1<\mu_2$, the above boundary limit inequality is never true, while whether it is valid for domains on which $\kappa_1=\mu_2$ remains open.

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