Home Mathematics Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
Article
Licensed
Unlicensed Requires Authentication

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay

  • Xavier Cabré EMAIL logo , Mouhamed Moustapha Fall , Joan Solà-Morales and Tobias Weth
Published/Copyright: April 16, 2016

Abstract

We are concerned with hypersurfaces of N with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in N with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in 2 with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

Funding source: MINECO

Award Identifier / Grant number: MTM2011-27739-C04-01

Funding source: MINECO

Award Identifier / Grant number: MTM2014-52402-C3-1-P

Funding statement: The first and third authors are supported by MINECO grants MTM2011-27739-C04-01 and MTM2014-52402-C3-1-P, and they are part of the Catalan research group 2014 SGR 1083. The second author’s work is supported by the Alexander von Humboldt foundation.

References

[1] N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim. 35 (2014), 793–815. 10.1080/01630563.2014.901837Search in Google Scholar

[2] A. D. Alexandrov, Uniqueness theorems for surfaces in the large. V (in Russian), Vestnik Leningrad. Univ. 13 (1958), 27–34; translation in Amer. Math. Soc. Transl. (2) 21 (1962), 412–416. Search in Google Scholar

[3] A. Ambrosetti and G. Prodi, A primer of nonlinear analysis, Cambridge Stud. Adv. Math. 34, Cambridge University Press, Cambridge 1995. Search in Google Scholar

[4] 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), 609–639. 10.2422/2036-2145.201202_007Search in Google Scholar

[5] X. Cabré, M. Fall and T. Weth, Delaunay hypersurfaces with constant nonlocal mean curvature, preprint (2016), http://arxiv.org/abs/1602.02623. 10.1016/j.matpur.2017.07.005Search in Google Scholar

[6] X. Cabré, A. Mas and J. Solà-Morales, Periodic solutions of nonlinear dispersive and fractional elliptic equations, forthcoming. Search in Google Scholar

[7] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111–1144. 10.1002/cpa.20331Search in Google Scholar

[8] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011), 203–240. 10.1007/s00526-010-0359-6Search in Google Scholar

[9] L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843–871. 10.1016/j.aim.2013.08.007Search in Google Scholar

[10] G. Ciraolo, A. Figalli, F. Maggi and M. Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. reine angew. Math. (2016), 10.1515/crelle-2015-0088. 10.1515/crelle-2015-0088Search in Google Scholar

[11] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. 10.1016/0022-1236(71)90015-2Search in Google Scholar

[12] A.-L. Dalibard and D. Gerard-Varet, On shape optimization problems involving the fractional laplacian, ESAIM Control Optim. Calc. Var. 19 (2013), 976–1013. 10.1051/cocv/2012041Search in Google Scholar

[13] J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Anal. 137 (2016), 357–380. 10.1016/j.na.2015.10.009Search in Google Scholar

[14] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858–892. 10.1016/j.jde.2013.10.006Search in Google Scholar

[15] J. Dávila, M. del Pino and J. Wei, Nonlocal s-minimal surfaces and Lawson cones, preprint (2014), http://arxiv.org/abs/1402.4173. 10.4310/jdg/1525399218Search in Google Scholar

[16] C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl. (1) 6 (1841), 309–315. Search in Google Scholar

[17] S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in n, preprint (2015), http://arxiv.org/abs/1410.3076v4. Search in Google Scholar

[18] M. M. Fall and S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var. 21 (2015), no. 4, 924–938. 10.1051/cocv/2014048Search in Google Scholar

[19] 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

[20] A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. reine angew. Math. (2015), 10.1515/crelle-2015-0006. 10.1515/crelle-2015-0006Search in Google Scholar

[21] L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Math. 128, Cambridge University Press, Cambridge 2000. 10.1017/CBO9780511569203Search in Google Scholar

[22] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2012), 33–39. 10.1007/s00526-012-0539-7Search in Google Scholar

[23] F. Schlenk and P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602–632. 10.1016/j.aim.2011.10.001Search in Google Scholar

[24] P. Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc. Var. Partial Differential Equations 37 (2010), 329–344. 10.1007/s00526-009-0264-zSearch in Google Scholar

[25] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112. 10.1002/cpa.20153Search in Google Scholar

[26] E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math. 81 (2013), 1–23. 10.1007/s00032-013-0199-xSearch in Google Scholar

Received: 2015-08-24
Revised: 2015-12-14
Published Online: 2016-04-16
Published in Print: 2018-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2015-0117/html
Scroll to top button