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Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

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Abstract

Given an elliptic integrand of class \(\mathscr {C}^{2,\alpha }\), we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

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Notes

  1. As in [18, 1.10] the symbol \(\bigodot ^2 X\) denotes the vectorspace of bilinear maps of the type \(X \times X \rightarrow \mathbf {R}^{}\).

  2. As in [18, 1.7.4] we write \(\mathbf {O}^*({n},{k})\) for the set of \(\alpha \in {{\,\mathrm{Hom}\,}}( \mathbf {R}^{n}, \mathbf {R}^{k})\) such that \(\alpha ^* \circ \alpha = {({{\,\mathrm{im}\,}}\alpha ^*)}_\natural \) and \(\alpha \circ \alpha ^* = \mathrm {id}_{ \mathbf {R}^{k}}\).

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Acknowledgements

The first author has been partially supported by the NSF DMS Grant No. 1906451. The second author was supported by the National Science Centre Poland grant no. 2016/23/D/ST1/01084.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD, 20742, USA

    Antonio De Rosa

  2. Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097, Warszawa, Poland

    Sławomir Kolasiński

  3. Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159, Augsburg, Germany

    Mario Santilli

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  1. Antonio De Rosa
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  2. Sławomir Kolasiński
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  3. Mario Santilli
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Correspondence to Antonio De Rosa.

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Communicated by I. Fonseca

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De Rosa, A., Kolasiński, S. & Santilli, M. Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets. Arch Rational Mech Anal 238, 1157–1198 (2020). https://doi.org/10.1007/s00205-020-01562-y

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