欧几里得球上 $L^2$-Poincaré 不等式与高斯 Poincaré 不等式的尖锐 $L^2$-稳定性及梯度稳定性层级
Sharp $L^2$-stability and gradient stability hierarchies of the $L^2$-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
- School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纽芬兰纪念大学格伦菲尔夫学院科学与环境学院）
- Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究欧几里得球和高斯测度下尖锐 $L^2$-Poincaré 不等式的稳定性，建立尖锐 $L^2$ 与梯度稳定性估计及稳定性层级，给出最优常数与等式刻画，并揭示谱间隙结构。
AI中文摘要：
我们研究欧几里得球上以及关于高斯测度的尖锐 $L^2$-Poincaré 不等式，并关注其稳定性。在两种情形下，尖锐常数是自伴算子的第一个非零特征值，而优化子集合是一个有限维线性空间。利用谱分解，我们建立了尖锐的 $L^2$-稳定性估计和尖锐的梯度稳定性估计，以及两种范数下稳定性不等式的稳定性，并给出了显式的最优常数和等式情形的完整刻画。然后我们证明这一过程可以继续。Poincaré 亏量等于到逐次稳定性不等式的优化子的递增空间的 $L^2$-距离的无穷和，也等于到同一空间的 $L^2$-梯度距离的无穷和。第一个和的系数是谱间隙 $\mu_j-\mu_{j-1}$，第二个和的系数是 $\mu_1$ 乘以倒数谱的间隙，即 $\mu_1\left(\mu_{j-1}^{-1}-\mu_j^{-1}\right)$。在欧几里得球上，所有常数都通过 Bessel 函数的根来表达，我们证明在所有维数下，前两个谱层由一次和二次 Neumann 模给出，而第三层在 $N=2,3$ 时由第一个径向模给出，在 $N\geq4$ 时由三次模给出。在高斯情形下，所有常数都是有理数，且 $L^2$-和的所有系数都等于 $1$；我们证明这一性质刻画了算术谱。
英文摘要：
We study the sharp $L^2$-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp $L^2$-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of $L^2$-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of $L^2$-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps $μ_j-μ_{j-1}$, and the coefficients of the second sum are $μ_1$ times the gaps of the reciprocal spectrum, $μ_1\left(μ_{j-1}^{-1}-μ_j^{-1}\right)$. On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when $N=2,3$ and by the degree three modes when $N\geq4$. In the Gaussian case, all the constants are rational numbers and the $L^2$-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.
