Some Regular and Non Regular Variational Problems

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete / 3. Folge. A Series of Modern Surveys in Mathematics book series (MATHE3, volume 38)


In this chapter we shall extend some of the methods and results of Ch. 4 to more general situations. In Sec. 5.1 we deal with Oseen-Frank energy for liquid crystals developing a theory which parallels the ones for harmonic maps from B 3 into S 2. Less complete is the theory for energy minimizing maps from a generic manifold X into S 2 and from a 2-dimensional manifold into a generic manifold y we develop respectively in Sec. 5.2 and Sec. 5.3. In the final Sec. 5.4 we introduce the notion of (n, p)-currents and develop a homological theory of the Dirichlet integral in the non regular case. Further results and question are stated in the Notes, Sec. 5.5.


Homotopy Class Homology Class Coarea Formula Rectifiable Current Dirichlet Energy 
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© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità di FirenzeFirenzeItaly
  3. 3.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Čzech Republic

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