Necessary and sufficient conditions for rigidity of Pólya–Szegö inequality under Schwarz symmetrisation
摘要
Pólya–Szegö inequality states that integral functionals, whose integrand is a convex function of the gradient, do not increase under Schwarz symmetrisation. In a seminal paper, Brothers and Ziemer gave sufficient conditions for the uniqueness (up to translations) of the extremal of the inequality. In this note we show that these conditions are also necessary.