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| Isoperimetric, Sobolev, and Eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method: A Survey |
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Citation： |
Xavier CABR'E.Isoperimetric, Sobolev, and Eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method: A Survey[J].Chinese Annals of Mathematics B,2017,38(1):201~214 |
| Page view： 971
Net amount： 787 |
Authors： |
Xavier CABR'E; |
Foundation： |
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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| 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. |
Keywords： |
Isoperimetric inequalities, Principal eigenvalue, Wulff shapes, ABPestimate |
Classification： |
28A75, 35P15, 35A23, 49Q20 |
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