带有径向对数凹测度的Neumann特征值的等周不等式
An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures
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中文总结 AI 辅助
该研究针对带径向对数凹测度的空间中原点对称Lipschitz区域，证明了Witten-Laplacian前n个非零Neumann特征值调和平均的尖锐等周不等式，扩展了此前的相关结果。
中文摘要 AI 辅助
我们针对空间中原点对称Lipschitz区域上Witten-Laplacian的前n个非零Neumann特征值的调和平均，证明了一个尖锐的等周不等式，该区域带有径向对数凹测度。主要创新点在于，我们在一般径向对数凹测度下建立了尖锐的调和平均不等式，且未要求权函数非增，这扩展了此前仅适用于特定或更受限加权情形的结果。证明依赖于测地球上第一特征函数的精细分析、由径向权的凸性条件导出的单调性性质，以及一个矩阵迹不等式。
英文摘要
We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.
