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Regularity of codimension-1 minimizing currents under minimal assumptions on the integrand. (English) Zbl 1386.49060

Summary: 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’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 \(C^{1,\alpha}\).

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53C65 Integral geometry
58A25 Currents in global analysis
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