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

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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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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028, Barcelona, Spain

    Xavier Cabré

  2. ICREA, Pg. Lluis Companys 23, 08010, Barcelona, Spain

    Xavier Cabré

Authors
  1. Xavier Cabré
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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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  • DOI: https://doi.org/10.1007/s11401-016-1067-0

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