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


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

2000 MR Subject Classification

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