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Regularity for graphs with bounded anisotropic mean curvature

Abstract

We prove that \(m\)-dimensional Lipschitz graphs with anisotropic mean curvature bounded in \(L^p\), \(p>m\), are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a \(C^{1,1}\) neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a \(C^{1,1}\) (resp. \(C^1\)) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

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Notes

  1. In fact, as written at the end of the proof of [8,  Proposition 6.8], \(C_1\) can be taken to be exactly \(\Vert H'\Vert _{L^p}\).

  2. More generally, one can identify \({\mathbb {G}}(N,m)\) with the space of simple \(m\)-vectors of \({\mathbb {R}}^N\), see [21,  Sect. 2.1].

  3. The computation in [23] is actually carried out for currents, but the evenness of \({\mathcal {G}}\) allows us to immediately extend it to varifolds.

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Acknowledgements

A. De Rosa has been partially supported by the NSF DMS Grant No. 1906451, the NSF DMS Grant No. 2112311, and the NSF DMS CAREER Award No. 2143124. R. Tione has been supported by the SNF Grant 182565. Most of this work was completed while R. Tione was affiliated with the University of Zurich and then with EPFL. Both authors wish to thank the anonymous referees for having provided useful suggestions to improve our first version of the paper.

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De Rosa, A., Tione, R. Regularity for graphs with bounded anisotropic mean curvature. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01129-6

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Mathematics Subject Classification

  • 49Q05
  • 49Q20
  • 53A10
  • 35D30