Calculus of Variations and Geometric Measure Theory

C. De Lellis - E. Spadaro

Regularity of area-minimizing currents I: $L^p$ gradient estimates

created by delellis on 02 Jul 2013
modified on 24 May 2017


Published Paper

Inserted: 2 jul 2013
Last Updated: 24 may 2017

Journal: Geom. Funct. Anal.
Volume: 24
Pages: 1831–1884
Year: 2014


In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area mi\-ni\-mi\-zing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an area minimizing current, together with several statements concerning approximations with Lipschitz multiple valued graphs. Our new a priori estimate is an higher integrability type result, which has a counterpart in the theory of Dir-minimizing multiple valued functions and plays a key role in estimating the accuracy of the Lipschitz approximations.