发表机构
University of Science and Technology of China; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Università degli Studi di Firenze（中国科学技术大学; 中国科学院数学与系统科学研究院; 佛罗伦萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了凸域上2-Hessian特征函数的严格对数凹性与扭转问题解的1/2-凸性，由此建立Brunn--Minkowski不等式，并构造反例说明该性质对3-Hessian方程失效。
AI 中文摘要
我们证明了在$\mathbb{R}^{n}$中光滑有界一致凸域上，2-Hessian方程的正第一特征函数$-u$的严格对数凹性，以及相应扭转问题解的严格$1/2$-凸性。作为应用，我们建立了相关的Brunn--Minkowski不等式。我们还通过构造四维空间中一个光滑一致凸域，其容许零边界解具有非凸子水平集，表明这种变换凸性现象对3-Hessian方程不成立。
英文摘要
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \(3\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.
CommentsThe conclusion of this article covers the original article(arxiv2606.22847)
