Some Regular and Non Regular Variational Problems
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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.
KeywordsHomotopy Class Homology Class Coarea Formula Rectifiable Current Dirichlet Energy
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