Abstract For any fixed integer D > 1 {D>1} we show that there exists M ∈ [ e - 1 , 1 ] {M\in[e^{-1},1]} such that for any open, bounded, convex domain Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on ( 1 , ∞ ) {(1,\infty)} . Moreover, for any real number s > M {s>M} there exists an open, bounded, convex domain Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of p ∈ ( 1 , ∞ ) {p\in(1,\infty)} .

中文翻译：

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关于 p-Laplacian 主频率的单调性

摘要 对于任何固定整数 D > 1 {D>1} 我们证明存在 M ∈ [ e - 1 , 1 ] {M\in[e^{-1},1]} 使得对于任何开、有界、凸域 Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} 具有平滑边界，其中到 Ω 边界的距离函数的最大值小于或等于 M，主频率Ω 上的 p-Laplacian 是 p on ( 1 , ∞ ) {(1,\infty)} 的增函数。此外，对于任何实数 s > M {s>M} 存在一个开放的、有界的、凸域 Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} 具有平滑边界，其具有到 Ω 边界的距离函数的最大值等于 s，使得 p-拉普拉斯算子的主频率不是 p ∈ ( 1 , ∞ ) {p\in(1,\infty)} 的单调函数。