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

作     者：Xavier CABRe 

作者机构：Universitat Politecnica de CatalunyaDepartament de MatemktiquesDiagonal 64708028 BarcelonaSpain ICREAPg.Lluis Companys 2308010 BarcelonaSpain. 

出 版 物：《Chinese Annals of Mathematics,Series B》 (数学年刊（B辑英文版）)

年 卷 期：2017年第38卷第1期

页      面：201-214页

核心收录：

学科分类：07[理学] 

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

主　　题：Isoperimetric inequalities Principal eigenvalue Wulff shapes ABP estimate 

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.