The symmetric (log-)epiperimetric inequality and a decay-growth estimate

2023/04/21 by Nick Edelen, Edelen, Nick, Luca Spolaor +3 · 1 citation
Computer Science · Decision Sciences · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Fatigue and fracture mechanics #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2304.11129

openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: as the radius increases energy decays while negative and grows while positive. One can view the symmetric epiperimetric inequality as giving a log-convexity of energy, analogous to the 3-annulus lemma or frequency formula. We establish the symmetric epiperimetric inequality for some free-boundary problems and almost-minimizing currents, and give some applications including a ``propagation of graphicality'' estimate, uniqueness of blow-downs at infinity, and a local Liouville-type theorem.

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