Calculus of Variations and Geometric Measure Theory

A. Figalli

Regularity of codimension-1 minimizing currents under minimal assumptions on the integrand

created by figalli on 01 Apr 2016


Accepted Paper

Inserted: 1 apr 2016
Last Updated: 1 apr 2016

Journal: J. Diff. Geom.
Year: 2016


In this paper we investigate the regularity theory of codimension-$1$ integer rectifiable currents that (almost)-minimize parametric elliptic functionals. While in the non-parametric case it follows by De Giorgi-Nash Theorem that $C^{1,1}$ regularity of the integrand is enough to prove $C^{1,\alpha}$ regularity of minimizers, the present regularity theory for parametric functionals assume the integrand to be at least of class $C^2$. In this paper we fill this gap by proving that $C^{1,1}$ regularity is enough to show that flat almost-minimizing currents are $C^{1,\alpha}$. As a corollary, we also show that the singular set has codimension greater than $2$.

Besides the result ``per se'', of particular interest we believe to be the approach used here: instead of showing that the standard excess function decays geometrically around every point, we construct a new excess with respect to graphs minimizing the non-parametric functional and we prove that if this excess is sufficiently small at some radius $R$, then it is identically zero at scale $R/2$. This implies that our current coincides with a minimizing graph there, hence it is of class $C^{1,\alpha}$.