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
As in [18, 1.10] the symbol \(\bigodot ^2 X\) denotes the vectorspace of bilinear maps of the type \(X \times X \rightarrow \mathbf {R}^{}\).
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 \(\alpha \circ \alpha ^* = \mathrm {id}_{ \mathbf {R}^{k}}\).
References
Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V. Vestnik Leningrad. Univ 13(19), 5–8, 1958
Allard, W.K.: An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled. Proceedings of Symposia in Pure Mathematics. Geometric Measure Theory and the Calculus of Variations, 44, 1986.
Allard, W.K.: On the first variation of a varifold. Ann. Math. 2(95), 417–491, 1972
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York 2000
Brothers, J.E., Morgan, F.: The isoperimetric theorem for general integrands. Michigan Math. J. 41(3), 419–431, 1994
Delgadino, M.G., Maggi, F., Mihaila, C., Neumayer , R.: Bubbling with \(L^2\)-almost constant mean curvature and an Alexandrov-type theorem for crystals. Arch. Rat. Mech. Anal. 230(3), 1131–1177, 2018
Delgadino, M.G., Maggi, F.: Alexandrov’s theorem revisited. Version 1 ofArxiv: 1711.07690v1, 2017.
Delgadino, M.G., Maggi, F.: Alexandrov’s theorem revisited. Anal. PDE 12(6), 1613–1642, 2019
De Philippis, G., De Rosa, A., Ghiraldin, F.: Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies. Comm. Pure Appl. Math. 71(6), 1123–1148, 2018
De Philippis, G., De Rosa, A., Ghiraldin, F.: Existence results for minimizers of parametric elliptic functionals. J. Geom. Anal. 30(2), 1450–1465, 2020
De Philippis, G., De Rosa, A., Hirsch, J.: The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete Contin. Dyn. Syst. - A. 39(12), 7031–7056, 2019
De Philippis, G., Maggi, F.: Regularity of free boundaries in anisotropic capillarity problems and the validity of Young’s law. Arch. Rat. Mech. Anal. 216(2), 473–568, 2015
De Rosa, A.: Minimization of anisotropic energies in classes of rectifiable varifolds. SIAM J. Math. Anal. 50(1), 162–181, 2018
De Rosa, A., Gioffrè, S.: Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces. arXiv e-prints, page arXiv:1803.09118, Mar 2018.
De Rosa, A., Gioffrè, S.: Quantitative stability for anisotropic nearly umbilical hypersurfaces. J. Geom. Anal. 29(3), 2318–2346, 2019
De Rosa, A., Kolasiński, S.: Equivalence of the ellipticity conditions for geometric variational problems. Commun. Pure Appl. Math. 2020. https://doi.org/10.1002/cpa.21890
Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491, 1959
Federer, H. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969D).
Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119(1–2), 125–136, 1991
Giga, Y.: Surface evolution equations: a level set method. Hokkaido Univ. Tech. Rep. Ser. Math. 71, 1, 2002
Giga, Y., Zhai, J.: Uniqueness of constant weakly anisotropic mean curvature immersion of the sphere \(S^2\) in \({\mathbb{R}}^3\). Adv. Differ. Equ. 14(7–8), 601–619, 2009
He, Y., Li, H., Ma, H., Ge, J.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. 58(2), 853–868, 2009
Heveling, M., Hug, D., Last, G.: Does polynomial parallel volume imply convexity? Math. Ann. 328(3), 469–479, 2004
Hug, D., Last, G., Weil, W.: A local Steiner-type formula for general closed sets and applications. Math. Z. 246(1–2), 237–272, 2004
Koiso, M.: Uniqueness of stable closed non-smooth hypersurfaces with constant anisotropic mean curvature. arXiv e-prints, page arXiv:1903.03951, 2019.
Koiso, M., Palmer, B.: Anisotropic umbilic points and Hopf’s theorem for surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J. 59(1), 79–90, 2010
Lang, S.: Linear Algebra. Undergraduate Texts in Mathematics., 3rd edn. Springer, New York 1987
Maggi, F.: Critical and almost-critical points in isoperimetric problems. Oberwolfach Rep. 35, 34–37, 2018
Menne, U., Santilli, M.: A geometric second-order-rectifiable stratification for closed subsets of Euclidean space. Annali Scuola Normale Superiore-Classw di Scienze 19(3):1185–1198
Menne, U.: Pointwise differentiability of higher order for sets. Ann. Global Anal. Geom. 55(3), 591–621, 2019
Milman, V.D., Schechtman, Gideon. Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics. Springer, Berlin, 1986. With an appendix by M. Gromov.
Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In Differential geometry, volume 52 of Pitman Monogr. Surveys Pure Appl. Math., pp. 279–296. Longman Sci. Tech., Harlow, 1991.
Morgan, F.: Planar Wulff shape is unique equilibrium. Proc. Amer. Math. Soc. 133(3), 809–813, 2005
Palmer, B.: Stability of the Wulff shape. Proc. Amer. Math. Soc. 126(12), 3661–3667, 1998
Palmer, B.: Stable closed equilibria for anisotropic surface energies: surfaces with edges. J. Geom. Mech. 4(1), 89–97, 2012
Tyrrell Rockafellar, R.: Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, 1970.
Santilli, M.: Fine properties of the curvature of arbitrary closed sets. Ann. Mat. Pura Appl. (4), 199, (2020), no. 4, 1431–1456.
Santilli, M.: Rectifiability and approximate differentiability of higher order for sets. Indiana Univ. Math. J. 68, 1013–1046, 2019
Santilli, M.: The Heintze-Karcher inequality for sets of finite perimeter and bounded mean curvature. Version 1 ofarXiv:1908.05952v1, Aug 2019.
Santilli, M.: Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature. Bull. Math. Sci. 10(1), 2050008, 24, 2020.
Schätzle, R.: Quadratic tilt-excess decay and strong maximum principle for varifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3(1):171–231, 2004.
Taylor, J.E.: Existence and structure of solutions to a class of nonelliptic variational problems. In Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), pp. 499–508. 1974.
Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In Differential geometry (Proceedings of Symposium, Pure. Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 419–427, 1975.
Wulff, G.: Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449–530, 1901
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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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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DOI: https://doi.org/10.1007/s00205-020-01562-y