Reilly type inequality for the first eigenvalue of the L_{r; F} operator

2011/12/09 by Yijun He, He, Yijun
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1112.2236

openalex publication_date 2011/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a positive function F on \mathbb Sⁿ which satisfies a convexity condition, for 1≤ r≤ n, we define for hypersurfaces in ℝⁿ⁺¹ the r-th anisotropic mean curvature function H_{r; F}, a generalization of the usual r-th mean curvature function. We also define L_{r; F} operator, the linearized operator of the r-th anisotropic mean curvature, which is a generalization of the usual L_r operator for hypersurfaces in the Euclidean space \mathbb Rⁿ⁺¹. The Reilly type inequalities for the first eigenvalue of the L_{r; F} operator have been proved.

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