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arXiv · 2608.17693

A Unified Quermassintegral Approach to Quasilinear Heat Dispersion and Loss

Xiaoshang Jin·Jie Xiao
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Abstract

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.

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Xiaoshang Jin, Jie Xiao. 2026-08-18. A Unified Quermassintegral Approach to Quasilinear Heat Dispersion and Loss. https://arxiv.org/abs/2608.17693

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