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

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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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References

  1. Ambrosio L., Dal Maso G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108(3), 691–702 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brezis H., Coron J.-M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37(2), 149–187 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brothers J.E., Morgan F.: The isoperimetric theorem for general integrands. Mich. Math. J. 41(3), 419–431 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brendle S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cicalese M., Leonardi G.P.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206(2), 617–643 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cianchi A., Salani P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345(4), 859–881 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Lellis C., Müller S.: Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom. 69(1), 75–110 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. De Lellis C., Müller S.: A C 0 estimate for nearly umbilical surfaces. Calc. Var. Partial Differ. Equ. 26(3), 283–296 (2006)

    Article  MATH  Google Scholar 

  10. De Rosa, A., Gioffré, S.: Quantitative stability for anisotropic nearly umbilical surfaces. 2017. Preprint available on arXiv

  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 1990

  12. Fonseca I., Müller S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119(1–2), 125–136 (1991)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Figalli A., Maggi F., Pratelli A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Fonseca I.: The Wulff theorem revisited. Proc. R. Soc. Lond. Ser. A 432(1884), 125–145 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Hebey,E.: Compactness and Stability for Nonlinear Elliptic Equations. European Mathematical Society (EMS), Zürich, Zurich Lectures in Advanced Mathematics 2014

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  22. Maggi F.: Sets of finite perimeter and geometric variational problems, vol. 135 Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  23. Morgan Frank: Planar Wulff shape is unique equilibrium. Proc.Am.Math Soc. 133(3), 809–813 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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 1991

  25. Ma H., Xiong C.: Hypersurfaces with constant anisotropic mean curvatures. J. Math. Sci. Univ. Tokyo 20(3), 335–347 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Perez, D.: On nearly umbilical surfaces. 2011. PhD. Thesis available at http://user.math.uzh.ch/delellis/uploads/media/Daniel.pdf

  27. Reilly R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26(3), 459–472 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ros A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3(3–4), 447–453 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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 1983

  30. Struwe M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187(4), 511–517 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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 1974

  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 1975

  34. Taylor J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568–588 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Xia Chao., Zhang Xiangwen.: ABP estimate and geometric inequalities. Commun. Anal. Geom. 25(3), 685–708 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK

    Matias G. Delgadino

  2. Department of Mathematics, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX, 78712, USA

    Francesco Maggi

  3. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Cornelia Mihaila

  4. Department of Mathematics, Northwestern University, Lunt Hall 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Robin Neumayer

Authors
  1. Matias G. Delgadino
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  2. Francesco Maggi
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  3. Cornelia Mihaila
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  4. Robin Neumayer
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Correspondence to Francesco Maggi.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by I. Fonseca

F. Maggi: On leave from the University of Texas at Austin.

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Delgadino, M.G., Maggi, F., Mihaila, C. et al. Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals. Arch Rational Mech Anal 230, 1131–1177 (2018). https://doi.org/10.1007/s00205-018-1267-8

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  • DOI: https://doi.org/10.1007/s00205-018-1267-8

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