We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue increases strictly with the apex angle $\theta$, while the first antisymmetric eigenvalue decreases strictly up to the equilateral triangle and increases strictly thereafter. \rev{The proof combines a slice-average estimate of directional energies with the explicit equilateral eigenfunction.} A quadratic correction across slices also gives a two-term expansion for every fixed positive spectral index $k$ as $\theta\downarrow0$, with relative correction $\theta^2/6$ and remainder $O_k(\theta^4)$.
We prove both inequalities in Laugesen and Siudeja's Conjecture~6.34 for the first two positive Neumann eigenvalues of triangles. The equilateral triangle uniquely maximizes their harmonic mean at fixed perimeter and their geometric mean at fixed area. Two-dimensional trial spaces reduce the proof to polynomial positiv...
We prove a sharp Szeg\"o-Weinberger-type inequality for balanced doubly connected domains in $\mathbb S^2$, $\mathbb R^2$ and $\mathbb H^2$. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the ar...
We prove P\'olya's inequality for every Neumann eigenvalue of every Euclidean circular annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. It is entirely analytic when the width is at most one quarter...
We prove that every Steklov isospectral family of compact smooth planar domains is compact in the $C^\infty$ topology, answering an open question of Colbois, Girouard, Gordon, and Sher. The proof has two main parts. First, a trace comparison principle for Dirichlet-to-Neumann operators yields monotonicity of negative S...
Let $\Omega\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operato...
We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=\pi/(2\...
S. Seto, Guo-Fang Wei, Yu-Sen Xia· 0 citations
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