Abstract
We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth.
Also, we extend to the nonlocal setting a famous theorem of De Giorgi [6] stating that the validity of Bernstein’s theorem in dimension
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1262411
Funding statement: Supported by NSF Grant DMS-1262411 (A. Figalli) and ERC Grant 277749 (E. Valdinoci).
Acknowledgements
It is a pleasure to thank Joaquim Serra for his keen comments on an earlier version of this paper.
References
[1] L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134 (2011), no. 3–4, 377–403. 10.1007/s00229-010-0399-4Search in Google Scholar
[2] B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609–639. 10.2422/2036-2145.201202_007Search in Google Scholar
[3] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. 10.1002/cpa.20331Search in Google Scholar
[4] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011), no. 1–2, 203–240. 10.1007/s00526-010-0359-6Search in Google Scholar
[5]
M. Cozzi,
On the variation of the fractional mean curvature under the effect of
[6] E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 19 (1965), 79–85. Search in Google Scholar
[7] A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys. 336 (2015), no. 1, 441–507. 10.1007/s00220-014-2244-1Search in Google Scholar
[8] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 33–39. 10.1007/s00526-012-0539-7Search in Google Scholar
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