Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime. (English) Zbl 1541.35478
Summary: The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields \(H^1 (S, T)\), where \(S\) and \(T\) are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface \(T\) is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.
MSC:
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 82D40 | Statistical mechanics of magnetic materials |
| 35J60 | Nonlinear elliptic equations |
| 35B07 | Axially symmetric solutions to PDEs |
| 35C08 | Soliton solutions |
| 35R09 | Integro-partial differential equations |
| 35A15 | Variational methods applied to PDEs |
| 78M30 | Variational methods applied to problems in optics and electromagnetic theory |
References:
| [1] | Golovaty, D.; Montero, J. A.; Sternberg, P., Dimension reduction for the Landau-de Gennes model on curved nematic thin films, J. Nonlinear Sci., 27, 6, 1905-1932, 2017 · Zbl 1382.35220 |
| [2] | Nitschke, I.; Nestler, M.; Praetorius, S.; Löwen, H.; Voigt, A., Nematic liquid crystals on curved surfaces: A thin film limit, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 474, 2214, Article 20170686 pp., 2018 · Zbl 1402.82021 |
| [3] | Curvilinear Micromagnetism, 2022, Springer International Publishing |
| [4] | Streubel, R.; Fischer, P.; Kronast, F.; Kravchuk, V. P.; Sheka, D. D.; Gaididei, Y.; Schmidt, O. G.; Makarov, D., Magnetism in curved geometries, J. Phys. D: Appl. Phys., 49, 36, Article 363001 pp., 2016 |
| [5] | Brown, W. F., Micromagnetics, 1963, Interscience Publishers, London · Zbl 0129.23401 |
| [6] | Hubert, A.; Schäfer, R., Magnetic Domains, 1998, Springer Berlin Heidelberg |
| [7] | Di Fratta, G.; Muratov, C. B.; Rybakov, F. N.; Slastikov, V. V., Variational principles of micromagnetics revisited, SIAM J. Math. Anal., 52, 4, 3580-3599, 2020 · Zbl 1446.35197 |
| [8] | Gioia, G.; James, R. D., Micromagnetics of very thin films, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 453, 1956, 213-223, 1997 |
| [9] | Carbou, G., Thin layers in micromagnetism, Math. Models Methods Appl. Sci., 11, 09, 1529-1546, 2001 · Zbl 1012.82031 |
| [10] | Di Fratta, G., Micromagnetics of curved thin films, Z. Angew. Math. Phys., 71, 4, 2020 · Zbl 1446.82087 |
| [11] | Di Fratta, G.; Slastikov, V., Curved thin-film limits of chiral Dirichlet energies, Nonlinear Anal., 234, Article 113303 pp., 2023 · Zbl 1518.49053 |
| [12] | Brown, W. F., Magnetostatic Principles in Ferromagnetism, 1962, North-Holland Publishing Company: North-Holland Publishing Company New York · Zbl 0116.23703 |
| [13] | Bertotti, G., Hysteresis in Magnetism, 1998, Elsevier |
| [14] | Eells, J.; Lemaire, L., Two Reports on Harmonic Maps, xii+216, 1995, World Scientific Publishing Co.: World Scientific Publishing Co. Inc., River Edge, NJ · Zbl 0836.58012 |
| [15] | Schoen, R.; Yau, S.-T., (Lectures on Harmonic Maps. Lectures on Harmonic Maps, Conference Proceedings and Lecture Notes in Geometry and Topology, II, 1997, International Press: International Press Cambridge, MA), vi+394 · Zbl 0886.53004 |
| [16] | Ball, J. M., Mathematics and liquid crystals, Mol. Cryst. Liq. Cryst., 647, 1, 1-27, 2017 |
| [17] | Virga, E. G., Variational Theories for Liquid Crystals, 2018, Chapman and Hall/CRC |
| [18] | Albeverio, S.; Jost, J.; Paycha, S.; Scarlatti, S., A Mathematical Introduction to String Theory, 1997, Cambridge University Press · Zbl 0882.53056 |
| [19] | Becker, K.; Becker, M.; Schwarz, J. H., String Theory and \(M\)-Theory, 2006, Cambridge University Press |
| [20] | Eells, J.; Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, 109-160, 1964 · Zbl 0122.40102 |
| [21] | Sandier, É., The symmetry of minimizing harmonic maps from a two dimensional domain to the sphere, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 5, 549-559, 1993 · Zbl 0797.35027 |
| [22] | Sandier, É.; Shafrir, I., On the symmetry of minimizing harmonic maps in \(N\) dimensions, Differential Integral Equations, 6, 6, 1531-1541, 1993 · Zbl 0780.58012 |
| [23] | Coron, J.-M.; Hélein, F., Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry, Compos. Math., 69, 2, 175-228, 1989 · Zbl 0686.58012 |
| [24] | Coron, J.-M.; Gulliver, R., Minimizing \(p\)-harmonic maps into spheres, J. Reine Angew. Math., 1989, 401, 82-100, 1989 · Zbl 0677.58021 |
| [25] | Baldes, A., Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Z., 185, 4, 505-516, 1984 · Zbl 0513.58021 |
| [26] | Hardt, R.; Lin, F.-H.; Poon, C.-C., Axially symmetric harmonic maps minimizing a relaxed energy, Comm. Pure Appl. Math., 45, 4, 417-459, 1992 · Zbl 0769.58013 |
| [27] | Hardt, R. M., Axially symmetric harmonic maps, (Nematics, 1991, Springer Netherlands), 179-187 · Zbl 0738.58014 |
| [28] | Martinazzi, L., A note on \(n\)-axially symmetric harmonic maps from \(B^3\) to \(\mathbb{S}^2\) minimizing the relaxed energy, J. Funct. Anal., 261, 10, 3099-3117, 2011 · Zbl 1237.58017 |
| [29] | Brezis, H., Symmetry in nonlinear PDE’s, (Differential Equations: La Pietra 1996 (Florence). Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math, vol. 65, 1999, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-12 · Zbl 0927.35038 |
| [30] | Pisante, A., Symmetry in nonlinear PDEs: Some open problems, J. Fixed Point Theory Appl., 15, 299-320, 2014 · Zbl 1311.35075 |
| [31] | Hélein, F., Harmonic Maps, Conservation Laws and Moving Frames, 2002, Cambridge University Press · Zbl 1010.58010 |
| [32] | Lin, F.; Wang, C., The Analysis of Harmonic Maps and their Heat Flows, xii+267, 2008, World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1203.58004 |
| [33] | Di Fratta, G.; Slastikov, V.; Zarnescu, A., On a sharp Poincaré-type inequality on the 2-sphere and its application in micromagnetics, SIAM J. Math. Anal., 51, 4, 3373-3387, 2019 · Zbl 1420.35012 |
| [34] | Melcher, C.; Sakellaris, Z. N., Curvature-stabilized skyrmions with angular momentum, Lett. Math. Phys., 109, 10, 2291-2304, 2019 · Zbl 1426.49051 |
| [35] | Fert, A.; Cros, V.; Sampaio, J., Skyrmions on the track, Nature Nanotechnol., 8, 3, 152-156, 2013 |
| [36] | Kravchuk, V. P.; Sheka, D. D.; Streubel, R.; Makarov, D.; Schmidt, O. G.; Gaididei, Y., Out-of-surface vortices in spherical shells, Phys. Rev. B, 85, 14, Article 144433 pp., 2012 |
| [37] | Kravchuk, V. P.; Rößler, U. K.; Volkov, O. M.; Sheka, D. D.; van den Brink, J.; Makarov, D.; Fuchs, H.; Fangohr, H.; Gaididei, Y., Topologically stable magnetization states on a spherical shell: Curvature-stabilized skyrmions, Phys. Rev. B, 94, 14, Article 144402 pp., 2016 |
| [38] | Di Fratta, G.; Fiorenza, A.; Slastikov, V., On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces, Math. Eng., 5, 3, 1-38, 2023 · Zbl 07817691 |
| [39] | Carbou, G.; Moussaoui, M.; Rachi, R., Stability of steady states in ferromagnetic rings, J. Math. Phys., 63, 3, Article 031508 pp., 2022 · Zbl 1507.35269 |
| [40] | Di Fratta, G.; Monteil, A.; Slastikov, V., Symmetry properties of minimizers of a perturbed Dirichlet energy with a boundary penalization, SIAM J. Math. Anal., 54, 3, 3636-3653, 2022 · Zbl 1494.35011 |
| [41] | Schoen, R.; Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18, 2, 1983 · Zbl 0547.58020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
