The method of moving planes: a quantitative approach

TL;DR

This paper reviews classical and recent quantitative results on the method of moving planes, highlighting its applications in symmetry and rigidity problems in PDEs and geometric analysis.

Contribution

It provides a comprehensive overview of both traditional and recent quantitative approaches to the method of moving planes.

Findings

Quantitative approximate symmetry results for overdetermined PDEs.

Applications to rigidity problems like Alexandrov's soap bubble theorem.

Enhanced understanding of symmetry in geometric analysis.

Abstract

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.