仿射变分泛函、等周问题与稳定性

中文摘要 AI 辅助

我们研究了基于Lutwak、Yang和Zhang的仿射$L^p$能量的三个仿射变分量：有界开集的仿射Poincaré--Sobolev常数$\Lambda^{\mathcal{A}}_{p,q}(\Omega)$、相关的仿射扭转刚度，以及紧集的仿射$p$-容量。我们证明椭球是相应等周不等式中的唯一极值集合，并且正体积紧集的仿射容量由求解仿射$p$-Laplace方程的位势达到。我们还证明了仿射Kohler-Jobin不等式：在具有给定仿射扭转刚度的集合中，只有椭球最小化$\Lambda^{\mathcal{A}}_{p,q}$。遵循Haddad、Jiménez和Montenegro处理仿射Sobolev不等式的方法，我们的方法依赖于$L_p$ Busemann--Petty质心不等式。通过证明其稳定性版本，我们获得了$\Lambda^{\mathcal{A}}_{p,q}$、仿射扭转刚度和仿射$p$-容量的等周不等式的定量形式，以及凸集上仿射Kohler-Jobin不等式的稳定性结果。

英文摘要

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré--Sobolev constants $Λ^{\mathcal{A}}_{p,q}(Ω)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $Λ^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann--Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $Λ^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

发表机构

• Universidad de Murcia（穆尔西亚大学）

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