Regularity of $\textrm{codimension-}1$ minimizing currents under minimal assumptions on the integrand

Abstract

In this paper, we investigate the regularity theory of $\textrm{codimension-}1$ integer rectifiable currents that (almost)-minimize parametric elliptic functionals. While in the non-parametric case it follows by De Giorgi–Nash’s 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 nonparametric 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 $C^{1,\alpha}$.