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Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals

  • Matias G. Delgadino
  • Francesco Maggi
  • Cornelia Mihaila
  • Robin Neumayer
Article
  • 1 Downloads

Abstract

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.

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Notes

Funding

RN supported by the NSF Graduate Research Fellowship under Grant DGE-1110007. FM, RN, andCMsupported by the NSF Grants DMS-1565354 and DMS-1361122.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ambrosio L., Dal Maso G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108(3), 691–702 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brezis H., Coron J.-M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37(2), 149–187 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brothers J.E., Morgan F.: The isoperimetric theorem for general integrands. Mich. Math. J. 41(3), 419–431 (1994)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brendle S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cicalese M., Leonardi G.P.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206(2), 617–643 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ciraolo G., Maggi F.: On the shape of compact hypersurfaces with almost-constant mean curvature. Comm. Pure Appl. Math. 70(4), 665–716 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cianchi A., Salani P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345(4), 859–881 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    De Lellis C., Müller S.: Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom. 69(1), 75–110 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    De Lellis C., Müller S.: A C 0 estimate for nearly umbilical surfaces. Calc. Var. Partial Differ. Equ. 26(3), 283–296 (2006)CrossRefMATHGoogle Scholar
  10. 10.
    De Rosa, A., Gioffré, S.: Quantitative stability for anisotropic nearly umbilical surfaces. 2017. Preprint available on arXivGoogle Scholar
  11. 11.
    Evans, L.C.: Weak convergence methods for nonlinear partial differential equations, volume 74 of CBMS Regional Conference Series in Mathematics . Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI 1990Google Scholar
  12. 12.
    Fonseca I., Müller S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119(1–2), 125–136 (1991)MathSciNetMATHGoogle Scholar
  13. 13.
    Figalli A., Maggi F.: On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201, 143–207 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Figalli A., Maggi F., Pratelli A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fonseca I.: The Wulff theorem revisited. Proc. R. Soc. Lond. Ser. A 432(1884), 125–145 (1991)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 2 nd edn. Springer, Berlin (1983)Google Scholar
  18. 18.
    Hebey,E.: Compactness and Stability for Nonlinear Elliptic Equations. European Mathematical Society (EMS), Zürich, Zurich Lectures in Advanced Mathematics 2014Google Scholar
  19. 19.
    He Y., Li H., Ma H., Ge J.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. 58(2), 853–868 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Krummel B., Maggi F.: Isoperimetry with upper mean curvature bounds and sharp stability estimates. Calc. Var. Partial Differ. Equ. 56(2), 56–53 (2017)Google Scholar
  21. 21.
    Leoni G., Morini M.: Necessary and sufficient conditions for thechain rule in \(W^{1,1}_{{\rm loc}}\) (\({\mathbb{R}^{N} ;\mathbb{R}^{d}}\)) and BVloc(\({\mathbb{R}^{N} ;\mathbb{R}^{d}}\)). J. Eur. Math. Soc. (JEMS) 9(2), 219–252 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Maggi F.: Sets of finite perimeter and geometric variational problems, vol. 135 Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  23. 23.
    Morgan Frank: Planar Wulff shape is unique equilibrium. Proc.Am.Math Soc. 133(3), 809–813 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Montiel, S.,Ros, A.:Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In: Differential geometry,volume 52 of Pitman Monogr. Surveys Pure Appl. Math. pp. 279–296. Longman Sci. Tech., Harlow 1991Google Scholar
  25. 25.
    Ma H., Xiong C.: Hypersurfaces with constant anisotropic mean curvatures. J. Math. Sci. Univ. Tokyo 20(3), 335–347 (2013)MathSciNetMATHGoogle Scholar
  26. 26.
    Perez, D.: On nearly umbilical surfaces. 2011. PhD. Thesis available at http://user.math.uzh.ch/delellis/uploads/media/Daniel.pdf
  27. 27.
    Reilly R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26(3), 459–472 (1977)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ros A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3(3–4), 447–453 (1987)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Simon,L.: Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis. Australian National University, Centre for Mathematical Analysis, Canberra 1983Google Scholar
  30. 30.
    Struwe M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187(4), 511–517 (1984)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Struwe, M.: Variational methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Applications to nonlinear partial differential equations and Hamiltonian systems. 3rd edn. Springer, Berlin (2000)Google Scholar
  32. 32.
    Taylor, J.E.: Existence and structure of solutions to a class of nonelliptic variational problems. In: Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), pp. 499–508. Academic Press, London 1974Google Scholar
  33. 33.
    Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In: Differential geometry (Proc. Sympos. Pure. Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1 ), pp. 419–427. Amer. Math. Soc., Providence, RI 1975Google Scholar
  34. 34.
    Taylor J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568–588 (1978)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wang G., Xia C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 199(1), 99–115 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xia Chao., Zhang Xiangwen.: ABP estimate and geometric inequalities. Commun. Anal. Geom. 25(3), 685–708 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA
  4. 4.Department of MathematicsNorthwestern UniversityEvanstonUSA

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