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

TL;DR

This survey explores the use of the ABP method to prove various inequalities, providing new proofs and insights into isoperimetric, Sobolev, and eigenvalue inequalities with applications to convex cones and weighted cases.

Contribution

The paper offers new and simplified proofs of key inequalities using the ABP technique, including classical and weighted Sobolev and isoperimetric inequalities.

Findings

New proof of a lower bound for the principal eigenvalue.

Simplified proofs of classical isoperimetric inequalities.

Extension to weighted Sobolev inequalities and convex cones.

Abstract

We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give 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 we also prove new Sobolev inequalities with weights which came up studying an open question raised by Haim Brezis.