Skip to main content

An Allard type regularity theorem for varifolds with a Hölder condition on the first variation

Abstract

We prove that Allard’s regularity theorem Allard (Ann Math 2(95):417–491, 1972) holds for rectifiable n-dimensional varifolds V assuming a weaker condition on the first variation. In particular, we do not assume the first variation to be bounded. Furthermore, we obtain a boundary regularity theorem in this setting thereby generalizing Allard’s boundary regularity theorem Allard (Ann Math 101(2):418–446, 1975) [cf. also (Bourni in Adv Calc Var 9(2):143–161, 2016)].

This is a preview of subscription content, access via your institution.

References

  1. Allard, W.K.: On the first variation of a varifold: boundary behavior. Ann. Math. 101(2), 418–446 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  2. Allard, W.K.: On the first variation of a varifold. Ann. Math. 2(95), 417–491 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  3. Ambrosio, L., Paolini, E.: Partial regularity for quasi minimizers of perimeter. Ricerche Mat. 48(suppl. 1), 167–186 (1999). Papers in memory of Ennio De Giorgi (Italian)

    MathSciNet  MATH  Google Scholar 

  4. Bourni, T.: Allard-type boundary regularity for \(C^{1,\alpha }\) boundaries. Adv. Calc. Var. 9(2), 143–161 (2016)

  5. Simon, L.: Lectures on geometric measure theory. In Proceedings of the Center for Mathematical Analysis, volume 3, Canberra, Australia, Australian National University (1983)

  6. Tamanini, I.: Regularity Results for Almost Minimal Oriented Hypersurfaces in \(\mathbb{R}^n\). Dipartimento di Matematica dell’Università di Lecce, Lecce (1984)

    MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank Ulrich Menne for useful conversations. We also thank Felix Schulze for useful comments on an earlier version of this paper. Part of this work was completed while the second-named author was financed by the grant ME 3816/2-1 of the DFG at the University of Potsdam.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Theodora Bourni.

Additional information

Communicated by C. De Lellis.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bourni, T., Volkmann, A. An Allard type regularity theorem for varifolds with a Hölder condition on the first variation. Calc. Var. 55, 46 (2016). https://doi.org/10.1007/s00526-016-0982-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-016-0982-y

Mathematics Subject Classification

  • 49Q15