Rigidity for perimeter inequality under spherical symmetrisation

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Calculus of Variations

Rigidity for perimeter inequality under spherical symmetrisation F. Cagnetti1 · M. Perugini1 · D. Stöger2 Received: 9 September 2019 / Accepted: 31 May 2020 / Published online: 4 August 2020 © The Author(s) 2020

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. Mathematics Subject Classification 49K21 · 49Q10 · 49Q20

1 Introduction 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. Perimeter inequalities under symmetrisation have been studied by many authors, see for instance [19,20] and the references therein. In general, we say that rigidity holds true for one of these inequalities if the set of extremals is trivial. The study of rigidity can have important applications to show that minimisers of variational problems (or solutions of PDEs) are symmetric. For instance, a crucial step in the proof of the Isoperimetric Inequality given by Ennio De Giorgi consists in showing rigidity of Steiner’s inequality (see, for instance, [21, Theorem 14.4]) for convex sets (see the proof of Theorem I in Section 4 in [15,16]). After De

Communicated by J. Ball.

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F. Cagnetti [email protected] M. Perugini [email protected] D. Stöger [email protected]

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Department of Mathematics, University of Sussex, Pevensey 2, Brighton BN1 9QH, UK

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Technische Universität München, Zentrum Mathematik - M15, Boltzmannstrasse 3, 85747 Garching, Germany

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F. Cagnetti et al.

Giorgi, an important contribution in the understanding of rigidity for Steiner’s inequality was given by Chlebík, Cianchi, and Fusco. 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 also given. Then, 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]. The anisotropic case has recently been considered in [25], where rigidity for Steiner’s inequality in the isotropic and anisotropic setting are shown to be equivalent, under suitable conditions. In the Gaussian setting, where the role of Steiner’s inequality is played by Ehrhard’s inequality (see [14, Section 4.1]), necessary and sufficient conditions for rigidity are given in [9], by making use of the notion of essential connectedness [9, Theorem 1.3]. Finally, in the smooth case,