Isoperimetric,Sobolev,and Eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method:A Survey  

作　　者：Xavier CABRe 

机构地区：[1]Universitat Politecnica de Catalunya,Departament de Matemktiques,Diagonal 647,08028 Barcelona,Spain [2]ICREA,Pg.Lluis Companys 23,08010 Barcelona,Spain.

出　　处：《Chinese Annals of Mathematics,Series B》2017年第1期201-214,共14页数学年刊（B辑英文版）

基　　金：supported by MINECO grant MTM2014-52402-C3-1-P

摘　　要：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.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.

关 键 词：Isoperimetric inequalities Principal eigenvalue Wulff shapes ABP estimate 

分 类 号：O[理学]