The Basic Elliptic Equations in an Equilateral Triangle

Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first $n$ eigenvalues of the Neumann Laplacian, when $n \geq 3$. The result fails for $n=2$, because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle.

Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

We address the question of determining the eigenvalues $λ\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results …

用于扩散和声学的等边三角形格子玻尔兹曼方案

A lattice Boltzmann scheme with equilateral triangles for diffusion and acoustics

提出基于等边三角形网格的D2T4格子玻尔兹曼方案，采用多松弛时间方法，通过考虑双点进行一致性分析，导出扩散和声学的等效偏微分方程，并在周期边界条件下验证收敛性。

Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelo…

等边三角形上 Schrödinger 方程的可观测性估计

Observability estimates for the Schrödinger equation on the equilateral triangle

本文研究等边三角形上 Schrödinger 方程的可观测性，借助 Pinsky 铺砌、奇性传播及带尖锐常数的 Strichartz 估计，在 Neumann 和 Dirichlet 条件下证明无需几何控制条件的新结果。

双曲平面中的拿破仑构造

Napoleonic Constructions in the Hyperbolic Plane

本文研究双曲平面中的拿破仑构造问题，证明双曲平面中仅等边三角形具备拿破仑性质，且任意三角形的迭代拿破仑化会逐渐趋近等边并收敛至单点，揭示双曲情形与欧氏、球面情形的显著差异。