Rigidity for perimeter inequality under spherical symmetrisation

Abstract

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.

Some isoperimetric inequalities in the plane with radial power weights

“…The main result of this Section is a tangential Vol'pert theorem given in Theorem 3.9. This result appears in [9] Theorem 3.7. This last is couched in the language of integer multiplicity rectifiable currents.…”

Isoperimetric residues and a mesoscale flatness criterion for hypersurfaces with bounded mean curvature

“…(5.22) see [CPS20]. Now, we can find a half-space J orthogonal to e n+1 and such that H n (J∩∂B r ) = H n (E j ∩∂B r ).…”

The $L^1$-relaxed area of the graph of the vortex map

“…This rearrangement is obtained slice by slice by spherical (two dimensional) symmetrization, a technique introduced first by Pòlya. We refer to [9] and references therein for a exhaustive description of the subject. Here we collect the main properties we will use in the sequel of the paper.…”

“…A proof of these properties is contained in [9,Theorem 1.4]. In particular, since U has finite perimeter, so is S(U ) and its perimeter cannot increase after symmetrization.…”