Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets
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Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets Antonio De Rosa , SŁawomir Kolasinski ´ & Mario Santilli Communicated by I. Fonseca
Abstract Given an elliptic integrand of class C 2,α , 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.
1. Introduction Overview The classical anisotropic isoperimetric problem (or Wulff problem) amounts to minimizing the anisotropic boundary energy among all sets of finite perimeter with prescribed volume. For all positive (continuous) integrands the solution is uniquely characterized up to translation by the Wulff shape, as proved by Taylor in [42] and [43]. Alternative proofs can be found in [5,19,31]. This isoperimetric shape was constructed by Wulff in [44] and plays a central role in crystallography. Instead of considering minima, a more subtle question asks to characterize critical points of the anisotropic boundary energy with prescribed volume. For integrands of class C 1 , this is equivalent to characterize sets of finite perimeter whose anisotropic mean curvature in the sense of varifolds is constant. For all convex integrands in R2 , Morgan proved in [33] that Wulff shapes are the only critical points among all planar regions with boundary given by a closed and connected rectifiable curve. To the best of our knowledge, the characterization in every dimension for smooth boundaries has been conjectured for the first time by Giga in [20] and Morgan in [33]. For smooth elliptic integrands, this has been positively answered in [21] for dimension 3, and in [22] for every dimension. In particular, He, Li, Ma and Ge proved in [22] that if F is a smooth elliptic integrand and is a set with smooth boundary and constant F-mean curvature (more generally constant higher
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order F-mean curvature), then is a Wulff shape. This is the anisotropic counterpart of the celebrated Alexandrov’s theorem [1]. Moreover, quantitative stability versions of this rigidity theorem have been showed in [6,14,15]. A weak version of the Alexandrov’s theorem for all convex integrands has been proved in [6]. Related results for immersed smooth hypersurfaces are [21,26,34]; for piecewise smooth hypersurfaces see [25,35]. In the non-smooth setting, Maggi has conjectured in [28, Conjecture] the characterization of the Wulff shapes among sets of finite perimeter: Conjecture. ([28]) For a positive convex integrand, Wulff shapes are the unique sets of finite perimeter and finite volume that are critical points of the anisotropic boundary energy at fixed volume. Since the integrand is assumed to be convex, but may fail to be C 1 , the notion of first variation and critical points are suitably defined using the convexity in time of the functional along any prescribed variational flow (see [28, p. 35–36]). Maggi specifies in [28] the significant interest f
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