Communications in Mathematical Physics

, Volume 131, Issue 3, pp 431–464 | Cite as

Variational problems on vector bundles

  • Jürg Fröhlich
  • Michael Struwe


A variety of problems in quantum physics and classical statistical mechanics, in particular the quantization of topological solitons and the statistical mechanics of defects in ordered media, are described. These problems can be studied within a semi-classical approximation, or with the help of low-temperature expansions, respectively. The calculation of the leading term in such expansions gives rise to variational problems for sections of vector bundles characterized by certain topological constraints. Examples of such problems are the quantization of kinks in the two-dimensional λϕ4-theory and the analysis of Bloch walls in a Landau-Ginzburg model of a threedimensional anisotropic ferromagnet. We state a general existence result for variational problems of this kind and develop regularity and decay estimates for solutions of the Landau-Ginzburg model describing Bloch walls with prescribed boundaries. For certain boundary configurations stability results are established. The relation between the minimizers of the Landau-Ginzburg model in a certain strong-coupling limit and minimal surfaces is pursued in some detail. An open question is whether, asymptotically, the stability of the limit (minimal) surface will imply the stability of the minimizers of the Landau-Ginzburg model.


Soliton Vector Bundle Statistical Mechanic Variational Problem Minimal Surface 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  • Michael Struwe
    • 1
  1. 1.Departments of Mathematics and PhysicsEidgenössische Technische HochschuleZürichSwitzerland

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