Regularity of area minimizing currents I: gradient Lp estimates

By Camillo De Lellis, Emanuele Spadaro. Geometric and Functional Analysis, 2013

99 citations recorded by the source corpus.

In a series of papers, including the present one, we give a new, shorter proof of Almgren’s partial regularity theorem for area minimizing 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 a 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c142e80ec5adae053d5778fb7ed8cecb5310ddba

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Regularity of area minimizing currents II: center manifold (2013, methodology): “Introduction In this second paper on the regularity of area minimizing integer rectifiable currents (we refer to the Foreword of [6] for the precise statement of the final theorem and on overview of its proof) we address one of the main steps in the analysis of the singularities, namely the…”
Multiple valued functions and integral currents (2013, background): “This is a fundamental idea for the study of the regularity of minimizing currents in the classical “single-vaued” case, and it also plays a fundamental role in the proof of Almgren’s partial regularity result (cf. [1, 5]).”