凸曲线与曲面的加权周长及p阶惯性矩
Weighted Perimeters and pth Moments of Inertia of Convex Curves and Surfaces
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中文总结 AI 辅助
该研究针对ℝⁿ中凸体的形状优化问题，证明了一般权重下极值的存在性，并在二维对称凸曲线中确定了特定退化针状构型为两类权重的最优解。
中文摘要 AI 辅助
我们研究了ℝⁿ中凸体之间的形状优化问题，该问题在标准周长约束下，最小化或最大化形如∫_∂Ω φ(|x|) dH^{n-1}(x)的加权周长。我们证明了任意维度下一般权重函数的极值存在性。在二维空间中，对于满足对称性假设的凸曲线，我们证明了退化针状构型(-a,a)×{0}⊂ℝ²是广泛权重族的最优解，包括p∈(0,2]时的|x|^p和α∈(0,1)时的|x|^{-α}。
英文摘要
We study a shape optimization problem among convex bodies in $\mathbb{R}^n$ that minimize or maximize weighted perimeters of the form $\int_{\partialΩ} ϕ(|x|) \, \mathrm{d} H^{n-1}(x)$ under a standard perimeter constraint. We prove the existence of extremals for general weight functions in any dimension. In dimension two, we prove that the degenerate needle configuration $(-a,a)\times \{0\} \subset \mathbb{R}^2$ is the optimizer for a wide family of weights, including $|x|^p$ for $p \in (0,2]$ and $|x|^{-α}$ for $α\in (0,1)$, among convex curves satisfying a symmetry assumption.
