Overdetermined problems - Part II

Abstract:

"I will first finish the proof of the symmetry result that the overdetermined Torsion problem is only solvable provided the domain is a ball. Next, I will revisit and prove the classical isoperimetric inequality using a method, attributed to Xavier Cabre, called the “lower contact set” which borrows ideas from the Alexandrov’s method of moving planes. I will then address the equality case and show that the equality could only hold provided the domain is a ball, and this is done by deriving a variant of the overdetermined Torsion problem which could only have a solution if the domain is a ball. I hope my description of the lower contact set would be somewhat clearer than that of Kesavan’s in his book “Symmetrization and Applications."