*Accepted Paper*

**Inserted:** 1 apr 2016

**Last Updated:** 1 apr 2016

**Journal:** J. Diff. Geom.

**Year:** 2016

**Abstract:**

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 $C^{1,\alpha}$.

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