Calculus of Variations and Geometric Measure Theory

N. Edelen - L. Spolaor - B. Velichkov

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

created by velichkov on 24 Apr 2023



Inserted: 24 apr 2023
Last Updated: 24 apr 2023

Year: 2023

ArXiv: 2304.11129 PDF


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.