arXiv subjects

Yingxin Sun

Publications and source records attributed to Yingxin Sun.

4 recordsLinked to original sources

arXiv2025

Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian

The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of this conjecture was later resolved by Ashbaugh and Benguria in the 1990s. For the Robin Laplacian, Payne and Schaefer (2001) formulated an analogous conjecture, positing that the ratio $\mu_2/\mu_1$ is also maximized by a disk for a range of the boundary parameter $\sigma$. This was later restated by Henrot in 2003. In this work, under some suitable conditions, we affirm this conjecture for all dimensions $N\geq2$ and for all $\sigma>0$. Furthermore, we prove that the maximum value of $\mu_2/\mu_1$ is strictly decreasing in $\sigma$ over the entire interval $(0,+\infty)$. Our result provides a positive answer to a variant of Yau's Problem 77: by measuring the ratio of the first two eigenfrequencies, one can determine whether an elastically supported drum is circular.

math.AP↗

arXiv2025

On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane

In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in $\mathbb{R}^2$. More precisely, assuming that a bounded uniformly convex domain $\Omega \subset \mathbb{R}^2$ has a connected boundary of class $C^{2,\epsilon}$ with $\epsilon \in (0,1)$, we show that if, for some nonzero constant $c_D$, the overdetermined elliptic problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad u = 0 \ \text{ on } \ \partial\Omega, \qquad \frac{\partial u}{\partial \nu} = c_{D} \ \text{ on } \ \partial\Omega \nonumber \end{equation} admits a nontrivial solution corresponding to a large eigenvalue $\alpha$, then the domain $\Omega$ must be a disk. Similarly, we establish that if a domain $\Omega \subset \mathbb{R}^2$ has a connected Lipschitz boundary and the problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \ \text{ on } \ \partial\Omega, \qquad u = c_{N} \ \text{ on } \ \partial\Omega \nonumber \end{equation} has a nontrivial solution corresponding to a large eigenvalue $\alpha$, then $\Omega$ is a disk as well.

math.AP↗

arXiv2025

Complete spectrum of the Robin eigenvalue problem on the ball

We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=\mu u\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+\alpha u=0 &\text{on}\,\, \partial B \end{array} \right. \end{equation*} on the unit ball of $\mathbb{R}^N$. We obtain the complete spectral structure of this problem. In particular, for $\alpha>0$, the first eigenvalue is $k_{\nu,1}^2$ and the second eigenvalue is $k_{\nu+1,1}^2$, where $k_{\nu+l,m}$ is the $m$th positive zero of $kJ_{\nu+l+1}(k)-(\alpha+l) J_{\nu+l}(k)$. Moreover, when $\alpha\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{\nu+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{\nu+l,1}$ denotes the unique zero of $\alpha I_{\nu+l}(k)+lI_{\nu+l}(k)+kI_{\nu+l+1}(k)$; while, for $\alpha=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.

math.AP↗

arXiv2023

Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder

Fix $R\in(0,1)$ and $N\geq2$, and let $\lambda_k$ be the $k$th eigenvalue of the radial Dirichlet problem on the annulus $A_R:=\{x\in\mathbb R^N:R<|x|<1\}$. For every $k\geq1$ we construct a smooth local family of rotationally symmetric, axially periodic perturbations of the hollow cylinder $A_R\times\mathbb R$ for which \begin{equation} -\Delta u=\lambda u\quad\hbox{in }\Omega,\qquad u=0\quad\hbox{on }\partial\Omega_{\mathrm{out}}\cup\partial\Omega_R,\qquad \partial_\nu u=\hbox{constant}\quad\hbox{on }\partial\Omega_{\mathrm{out}} \nonumber \end{equation} admits a nonradial periodic solution. The branch bifurcates from the $k$th radial eigenfunction. If $k\geq2$, the solution is sign-changing: it has precisely $k-1$ smooth nested nodal hypersurfaces and hence exactly $k$ nodal domains. We also give an explicit connected-strip analogue when $N=1$. The proof is based on a mean-free Dirichlet-to-Neumann operator. A Sturm--Liouville Weyl function yields a unique simple crossing in the first spectral gap, so that the Crandall--Rabinowitz theorem produces a smooth local branch. This gives a partially overdetermined construction on an unbounded hollow domain with disconnected complement; it lies outside the positivity and connected-complement hypotheses of the Berestycki--Caffarelli--Nirenberg conjecture.

math.AP↗