AI 中文总结
本文建立拟线性位势理论与凸几何分析的联系，引入凸导体的拟线性热扩散定律，证明固定平均宽度下闭球是该扩散及对应热损失的唯一最大化者，搭建了凸导体度量性质与拟线性椭圆算子变分分析的新桥梁。
AI 中文摘要
本文通过研究拟线性拉普拉斯算子与Quermassintegrals的相互作用，建立了拟线性位势理论与凸几何分析之间的基本联系。我们为凸导体引入了一种拟线性热扩散定律，并证明在所有具有固定平均宽度的凸导体中，闭球是该扩散的唯一最大化者。通过用Quermassintegrals明确刻画凸导体的拟线性热损失，我们不仅证明了数学物理背景下的等容容量不等式与等周不等式之间的形式等价性，还证明在所有具有固定平均宽度的凸导体中，闭球是该损失的唯一最大化者。这些结果为凸导体的度量性质与拟线性椭圆算子的变分分析之间搭建了一座新桥梁，为尖锐几何不等式及其极值情况提供了统一视角。
英文摘要
This paper establishes a fundamental connection between quasilinear potential theory and convex geometric analysis by investigating the interplay between the quasilinear Laplace operator and quermassintegrals. We introduce a quasilinear heat dispersion law for convex conductors and prove that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this dispersion. By characterizing the quasilinear heat loss of a convex conductor explicitly in terms of its quermassintegrals, we demonstrate not only a formal equivalence between the isocapacitary and isoperimetric inequalities in the setting of mathematical physics but also that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this loss. These results provide a novel bridge between the metric properties of convex conductors and the variational analysis of quasilinear elliptic operators, offering a unified perspective on sharp geometric inequalities and their extremal cases.
Comments24 pages
