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Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey

Abstract

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

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References

  1. [1]

    Berestycki, H. and Nirenberg, L., On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22, 1991, 1–37.

  2. [2]

    Berestycki, H., Nirenberg, L. and Varadhan, S. R. S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47, 1994, 47–92.

  3. [3]

    Besson, G., From isoperimetric inequalities to heat kernels via symmetrisation, Surveys in Differential Geometry, Vol. IX, 27–51, Surv. Differ. Geom., 9, Int. Press, Somerville, MA,2004.

  4. [4]

    Brezis, H., Is there failure of the inverse function theorem? Morse theory, minimax theory and their applications to nonlinear differential equations, Proc. Workshop held at the Chinese Acad. of Sciences, Beijing, 1999, 23–33, New Stud. Adv. Math., 1, Int. Press, Somerville, MA,2003.

  5. [5]

    Brezis, H. and Lions, P.-L., An estimate related to the strong maximum principle, Boll. Un. Mat. Ital. A, 17(5), 1980, 503–508.

  6. [6]

    Brezis, H. and Vázquez, J. L., Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10, 1997, 443–469.

  7. [7]

    Cabré, X., On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48, 1995, 539–570.

  8. [8]

    Cabré, X., Partial differential equations, geometry, and stochastic control (in Catalan), Butl. Soc. Catalana Mat., 15, 2000, 7–27.

  9. [9]

    Cabré, X., Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 8, 2002, 331–359.

  10. [10]

    Cabré, X., Elliptic PDEs in probability and geometry, symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., 20, 2008, 425–457.

  11. [11]

    Cabré, X., Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math., 63, 2010, 1362–1380.

  12. [12]

    Cabré, X., Cinti, E., Pratelli, A., et al., Quantitative isoperimetric inequalities with homogeneous weights, in preparation.

  13. [13]

    Cabré, X. and Ros-Oton, X., Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38, 2013, 135–154.

  14. [14]

    Cabré, X. and Ros-Oton, X., Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations, 255, 2013, 4312–4336.

  15. [15]

    Cabré, X., Ros-Oton, X. and Serra, J., Euclidean balls solve some isoperimetric problems with nonradial weights, C. R. Math. Acad. Sci. Paris, 350, 2012, 945–947.

  16. [16]

    Cabré, X., Ros-Oton, X. and Serra, J., Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc., 18, 2016, 2971–2998.

  17. [17]

    Cabré, X., Sanchón, M. and Spruck, J., A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst., Series A, 36, 2016, 601–609.

  18. [18]

    Caffarelli, L. A. and Cabré, X., Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.

  19. [19]

    Chavel, I., Riemannian Geometry: A Modern Introduction, 2nd Revised Edition, Cambridge University Press, Cambridge, 2006.

  20. [20]

    Cordero-Erausquin, C., Nazaret, B. and Villani, C., A mass transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182, 2004, 307–332.

  21. [21]

    Dinghas, A., Über einen geometrischen satz von Wulff für die gleichgewichtsform von kristallen, Zeitschrift für Kristallographie, 105, 1944, 304–314.

  22. [22]

    Druet, O., Isoperimetric inequalities on nonpositively curved spaces, Lecture Notes. http://math.arizona.edu/˜dido/presentations/Druet-Carthage.pdf

  23. [23]

    Fusco, N., The stability of the isoperimetric inequality, CNA Summer School, Carnegie Mellon University, Pittsburgh, 2013.

  24. [24]

    Gardner, R. J., The Brunn-Minkowski inequality, Bull. Amer. Math. Soc., 39, 2002, 355–405.

  25. [25]

    Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin-New York, 1983.

  26. [26]

    Gromov, M., Isoperimetric inequalities Riemannian manifolds, asymptotic theory of finite-dimensional normed spaces, Lecture Notes Math., 1200, Appendix I, 114–129, Springer-Verlag, Berlin-New York, 1986.

  27. [27]

    Hörmander, L., Linear Partial Differential Operators, Springer-Verlag, Berlin-New York, 1969.

  28. [28]

    Klimov, V. S., On the symmetrization of anisotropic integral functionals, Izv. Vyssh. Uchebn. Zaved. Mat., 99, 1999, 26–32 (in Russian); translation in Russian Math. (Iz. VUZ), 43, 1999, 23–29.

  29. [29]

    Lions, P.-L. and Pacella, F., Isoperimetric inequality for convex cones, Proc. Amer. Math. Soc., 109, 1990, 477–485.

  30. [30]

    Milman, E. and Rotem, L., Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, Adv. Math., 262, 2014, 867–908.

  31. [31]

    Osserman, R., The isoperimetric inequality, Bull. Amer. Math. Soc., 84 1978, 1182–1238.

  32. [32]

    Serra, J. and Teixidó, M., Isoperimetric inequality in Hadamard manifolds of dimension two via the ABP method, in preparation.

  33. [33]

    Taylor, J., Existence and structure of solutions to a class of nonelliptic variational problems, Symposia Mathematica, 14, 1974, 499–508.

  34. [34]

    Taylor, J., Unique structure of solutions to a class of nonelliptic variational problems, Proc. Symp. Pure Math., A. M. S., 27, 1975, 419–427.

  35. [35]

    Trudinger, N. S., Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 1994, 411–425.

  36. [36]

    Van Schaftingen, J., Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 2006, 539–565.

  37. [37]

    Wulff, G., Zur frage der geschwindigkeit des wachsturms und der auflösung der kristallflächen, Zeitschrift für Kristallographie, 34, 1901, 449–530.

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Author information

Correspondence to Xavier Cabré.

Additional information

Dedicated to Haïm Brezis, with great admiration

This work was supported by MINECO grant MTM2014-52402-C3-1-P. The author is part of the Catalan research group 2014 SGR 1083.

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Cabré, X. Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey. Chin. Ann. Math. Ser. B 38, 201–214 (2017). https://doi.org/10.1007/s11401-016-1067-0

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Keywords

  • Isoperimetric inequalities
  • Principal eigenvalue
  • Wulff shapes
  • ABP estimate

2000 MR Subject Classification

  • 28A75
  • 35P15
  • 35A23
  • 49Q20