Symmetrization with respect to mixed volumes
Symmetrization with respect to mixed volumes
复制标题
混合体积的对称性
DOI：
10.1016/j.aim.2021.107887
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发表时间：
2021
影响因子：
1.7
通讯作者：
Xia Chao
中科院分区：
文献类型：
--
作者：
Della Pietra Francesco;Gavitone Nunzia;Xia Chao
本文介绍了关于混合体积或各向异性曲率积分的一种新的对称化，它推广了Talenti [26]和Tso [28]关于quermass积分的对称化.我们展示了一个Pólya-SzegIgntype原则，这样的对称化-它减少了各向异性的海森积分的准凸函数。我们实现了这一点，通过系统的研究不变量的非对称矩阵的真实的特征值和高阶各向异性平均曲率的水平集，这可能是独立的兴趣。作为应用，我们建立了各向异性Hessian方程和各向异性Sobolev不等式的比较原理。
In this paper we introduce a new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti [26] and Tso [28]. We show a Pólya-Szegő type principle for such symmetrization — it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotropic Hessian equations and sharp anisotropic Sobolev inequalities.