A note on monotonicity and Bochner formulas in Carnot groups

2022/04/27 by Nicola Garofalo, Garofalo, Nicola · 1 citation
Engineering · Mathematics · #31C05 #35H10 #35R03 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2204.13073

openalex publication_date 2022/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we prove two monotonicity formulas for solutions of Δ_H f = c and Δ_H f - \p_t f = c in Carnot groups. Such formulas involve the right-invariant carré du champ of a function and they are false for the left-invariant one. The main results, Theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt-Caffarelli-Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an "almost monotoniccity" formula be possible. In the last section we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.

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