Symmetrization with respect to mixed volumes

2019/02/13 by Francesco Della Pietra, Della Pietra, Francesco, Nunzia Gavitone +3 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1902.04965

openalex publication_date 2019/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we introduce new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show a Pólya-Szego 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.

Citations

• Stability of hypersurfaces with constant (r+1)-th anisotropic mean curvature

Cited by

• General monotone formula for homogeneous k-Hessian equation in the exterior domain and its applications
• Stability and rigidity results of space-like hypersurface in the Minkowski space
• Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

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