The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

Guido De Philippis; Antonio De Rosa; Jonas Hirsch

doi:10.3934/dcds.2019243

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2019, Volume 39, Issue 12: 7031-7056. Doi: 10.3934/dcds.2019243

This issue Previous Article Remarks on some minimization problems associated with BV norms Next Article On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

1.
SISSA, Via Bonomea 265, Trieste, 34136, Italy
2.
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA
3.
Mathematisches Institut, Universität Leipzig, Augustus Platz 10, Leipzig, D04109, Germany

^* Corresponding author: Guido De Philippis
^* Corresponding author: Guido De Philippis

Received: January 2019

Revised: February 2019

Published online: December 2019

The work of G.D.P. is supported by the INDAM-grant "Geometric Variational Problems".

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Abstract

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

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Mathematics Subject Classification: Primary: 49Q05, 53A10; Secondary: 35D40.

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References

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