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.
Highlights
In this paper we study the perimeter inequality under spherical symmetrisation, giving necessary and sufficient conditions for the uniqueness, up to orthogonal transformations, of the extremals
Our first result shows that the spherical symmetrisation does not increase the perimeter, and gives some necessary conditions for equality cases
We observe that f ∨ and f ∧ are Borel functions that are defined at every point of Rn, with values in R ∪ {±∞}
Summary
In this paper we study the perimeter inequality under spherical symmetrisation, giving necessary and sufficient conditions for the uniqueness, up to orthogonal transformations, of the extremals. The study of rigidity can have important applications to show that minimisers of variational problems (or solutions of PDEs) are symmetric. In the seminal paper [11], the authors give sufficient conditions for rigidity which are much more general than convexity. After that, this result was extended to the case of higher codimensions in [2], where a quantitative version of Steiner’s inequality was given. Necessary and sufficient conditions for rigidity (in codimension 1) were given in [8], in the case where the distribution function is a Special Function of Bounded Variation with locally finite jump set [8, Theorem 1.29]. We direct the interested reader to [3,4,5,30] and the references therein for more information
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