Abstract
Anisotropic surface energies are a natural generalization of the perimeter functional that arise, for instance, in scaling limits for certain probabilistic models on lattices. We survey two recent results concerning isoperimetric problems with anisotropic surface energies. The first is joint work with Delgadino, Maggi, and Mihaila and provides a weak characterization of critical points in the anisotropic isoperimetric problem. The second is joint work with Choksi and Topaloglu and describes energy minimizers in an anisotropic variant of a model for atomic nuclei.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. V. Vestnik Leningrad. Univ. 13(19), 5–8 (1958)
Auffinger, A., Damron, M., Hanson, J.: 50 years of first-passage percolation, University Lecture Series, vol. 68. American Mathematical Society, Providence, RI (2017)
Bonacini, M., Cristoferi, R.: Local and global minimality results for a nonlocal isoperimetric problem on \(\mathbb{R}^{N}\). SIAM J. Math. Anal. 46(4), 2310–2349 (2014)
Brothers, J.E., Morgan, F.: The isoperimetric theorem for general integrands. Michigan Math. J. 41(3), 419–431 (1994). DOI https://doi.org/10.1307/mmj/1029005070. URL http://dx.doi.org.ezproxy.lib.utexas.edu/10.1307/mmj/1029005070
Cerf, R.: The Wulff crystal in Ising and percolation models, Lecture Notes in Mathematics, vol. 1878. Springer-Verlag, Berlin (2006). Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With a foreword by Jean Picard
Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59(3), 330–343 (2006). https://doi.org/10.1002/cpa.20116
Choksi, R., Neumayer, R., Topaloglu, I.: Anisotropic liquid drop models. Preprint available at arXiv:1810.08304
Choksi, R., Peletier, M.: Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal. 42(3), 1334–1370 (2010). DOI https://doi.org/10.1137/090764888. URL http://dx.doi.org/10.1137/090764888
Choksi, R., Peletier, M.: Small volume-fraction limit of the diblock copolymer problem: II. Diffuse-interface functional. SIAM J. Math. Anal. 43(2), 739–763 (2011). DOI https://doi.org/10.1137/10079330x. URL http://dx.doi.org/10.1137/10079330X
Cook, N.: Models of the Atomic Nucleus: Unification Through a Lattice of Nucleons. Springer Berlin Heidelberg (2010). URL https://books.google.com/books?id=CwRGogWF5-oC
Delgadino, M.G., Maggi, F., Mihaila, C., Neumayer, R.: Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals. Arch. Ration. Mech. Anal. 230(3), 1131–1177 (2018). DOI https://doi.org/10.1007/s00205-018-1267-8. URL https://doi.org.turing.library.northwestern.edu/10.1007/s00205-018-1267-8
Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys. 336-1, 441–507 (2015)
Figalli, A., Maggi, F.: On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201(1), 143–207 (2011). DOI https://doi.org/10.1007/s00205-010-0383-x. URL http://dx.doi.org/10.1007/s00205-010-0383-x
Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010). DOI https://doi.org/10.1007/s00222-010-0261-z. URL http://dx.doi.org/10.1007/s00222-010-0261-z
Fonseca, I.: The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432(1884), 125–145 (1991). DOI https://doi.org/10.1098/rspa.1991.0009. URL http://dx.doi.org.ezproxy.lib.utexas.edu/10.1098/rspa.1991.0009
Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119(1–2), 125–136 (1991). DOI https://doi.org/10.1017/s0308210500028365. URL http://dx.doi.org.ezproxy.lib.utexas.edu/10.1017/S0308210500028365
Fraenkel, L.E.: An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, vol. 128. Cambridge University Press, Cambridge (2000). URL https://doi.org/10.1017/CBO9780511569203
Frank, R.L., Lieb, E.H.: A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47(6), 4436–4450 (2015)
Gamow, G.: Mass defect curve and nuclear constitution. Proc. R. Soc. Lond. A 126(803), 632–644 (1930). DOI https://doi.org/10.1098/rspa.1930.0032. URL http://rspa.royalsocietypublishing.org/content/126/803/632
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). DOI https://doi.org/10.1512/iumj.2009.58.3515. URL http://dx.doi.org/10.1512/iumj.2009.58.3515
Julin, V.: Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J. 63(1), 77–89 (2014). URL https://doi.org/10.1512/iumj.2014.63.5185
Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing nonlocal term I: The planar case. Comm. Pure Appl. Math. 66(7), 1129–1162 (2013). URL https://doi.org/10.1002/cpa.21451
Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math. 67(12), 1974–1994 (2014). URL https://doi.org/10.1002/cpa.21479
Lu, G., Zhu, J.: An overdetermined problem in Riesz-potential and fractional Laplacian. Non-linear Anal. 75(6), 3036–3048 (2012). URL https://doi.org/10.1016/j.na.2011.11.036
Lu, J., Otto, F.: Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model. Comm. Pure Appl. Math. 67(10), 1605–1617 (2014). URL https://doi.org/10.1002/cpa.21477
Mackie, F.D.: Anisotropic surface tension and the liquid drop model. Nuclear Physics A 245(1), 61–86 (1975). DOI https://doi.org/10.1016/0375-9474(75)90082-2. URL http://www.sciencedirect.com/science/article/pii/0375947475900822
Maggi, F.: Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge (2012). URL https://doi.org/10.1017/CBO9781139108133. An introduction to geometric measure theory
Reichel, W.: Characterization of balls by Riesz-potentials. Ann. Mat. Pura Appl. (4) 188(2), 235–245 (2009). URL https://doi.org/10.1007/s10231-008-0073-6
Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3(3–4), 447–453 (1987)
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. Academic Press, London (1974)
Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In: Differential geometry (Proc. Sympos. Pure. Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 419–427. Amer. Math. Soc., Providence, R.I. (1975)
Wulff, G.: Zur frage der geschwindigkeit des wachsturms und der auflӧsungder kristallflächen. Z. Kristallogr. 34, 449–530 (1901)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Neumayer, R. (2020). On Minimizers and Critical Points for Anisotropic Isoperimetric Problems. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-38230-8_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38229-2
Online ISBN: 978-3-030-38230-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)