Abstract
The solutions of Skyrme's variational problem describe the structure of mesons in a field of weak energy. The problem consists in minimizing the corresponding energy among the functions from ℝ3 toS 3 which have a fixed “degree” without making any symmetry assumptions. We prove the existence of minima and study their properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adkins, G. S. Nappi, C. R., Witten, E.: Static properties of nucleons in the Skyrme model. Nucl. Phys.B228, 552–566 (1983)
Ball, J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.63, 337–403 (1977)
Brézis, H., Coron, J. M.: Large solutions for Harmonic maps inR 2. Commun. Math. Phys.92, 203–215 (1983)
Esteban, M. J.: An isoperimetric inequality inR 3. (To appear)
Esteban, M. J.: Existence of symmetric solutions for the Skyrme's problem. (To appear)
t'Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys.B72, 461–473 (1974)
t'Hooft, G.: A two-dimensional model for mesons. Nucl. Phys.B75, 461–470 (1974)
Kapitanski, L. B., Ladyzenskaia, O. A.: On the Coleman's principle concerning the stationary points of invariant functionals. Zap. Nauchn. Semin, LOMI127, 84–102 (1983)
Levy, P.: Theorie de l'addition des variables aléatoires. Paris: Gauthier-Villars 1954
Lions, P. L.: The Concentration-compactness principle in the calculus of variations: Part I.—Ann. IHP. Anal. Non Lin.1, 109–145 (1984); Part II.—Ann. IHP. Anal. Non Lin.1, 223–283 (1984)
Lions, P. L.: The Concentration-compactness principle in the calculus of variations. The limit case. Parts I and II: Part I.—Rev. Mat. Iber. I,1 145–200 (1985); Part II.—Rev. Mat. Iber. 1, 2, 45–121 (1985)
Murat, F.: Compacité par compensation: Conditions nécessaires et suffisantes de continuité faible sous une hypothèse de rang constant. Ann. Soc. Norm. Super. Pisa, IV, vol.VIII, 1, 69–102 (1981)
Nirenberg, L.: Topics in nonlinear functional analysis, N.Y.U. Lectures Notes, 1973–1974
Reshetnyak, Y. G.: Stability theorems for mappings with bounded excursions. Sib. Math. J.9, 499–512 (1968)
Skyrme, T. H. R.: A non-linear theory of strong interactions. Proc. Roy. Soc.A247, 260–278 (1958)
Skyrme, T. H. R.: A unifical model ofK and Π-mesons. Proc. Roy. Soc.A252, 236–245 (1951)
Skyrme, T. H. R.: A non-lineal field theory. Proc. Roy. Soc.A260, 127–138 (1961)
Tartar, L.: Compensated compactness and applications to partial differential equations. Non-linear analysis and mechanics: Heriot-Watt symposium, Vol. IV, Knops, R. J., (ed.), New York: Pitmann 1979
Taubes, C. H.: Monopoles and maps fromS 2 toS 2; the topology of the configuration space. Commun. Math. Phys.95, 345–391 (1984)
Taubes, C. H.: Min-Max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473–540 (1985)
Uhlenbeck, K. K.: Removeable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11–29 (1982)
Uhlenbeck, K. K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31–42 (1982)
Vinh Mau, G. S. Lacombe, M., Loiseau, B., Cottingham, W. N., Lisboa, P.: The static baryon-baryon potential in the SKYRME Model. Preprint
Witten, E.: Baryons in the 1/N expansion. Nucl. Phys.B160, 57–115 (1979)
Author information
Authors and Affiliations
Additional information
Communicated by L. Nirenbery
Rights and permissions
About this article
Cite this article
Esteban, M.J. A direct variational approach to Skyrme's model for meson fields. Commun.Math. Phys. 105, 571–591 (1986). https://doi.org/10.1007/BF01238934
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01238934