Lower Bounds for Generalized Ginzburg--Landau Functionals

We study properties of Ginzburg--Landau functionals I^U(·), defined for functions u∈ W1,n(U; Rⁿ), where U⊂ Rⁿ. In particular, we establish lower bounds relating the energy I^U(u) to the Brouwer degree of u, and we prove under additional hypotheses that the energy concentrates on a small number of small sets. As a consequence we deduce some compactness theorems. Such estimates are useful in studying Ginzburg--Landau-type PDEs associated with the functional I^U.