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)].
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References
Allard, W.K.: On the first variation of a varifold: boundary behavior. Ann. Math. 101(2), 418–446 (1975)
Allard, W.K.: On the first variation of a varifold. Ann. Math. 2(95), 417–491 (1972)
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Bourni, T.: Allard-type boundary regularity for \(C^{1,\alpha }\) boundaries. Adv. Calc. Var. 9(2), 143–161 (2016)
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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.
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Communicated by C. De Lellis.
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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
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DOI: https://doi.org/10.1007/s00526-016-0982-y