Lower Bounds for Generalized Ginzburg--Landau Functionals

Abstract

We study properties of Ginzburg--Landau functionals $I^\e_U(\cdot)$, defined for functions $u\in W^{1,n}(U; {\cal R}^n)$, where $U\subset {\cal R}^n$. In particular, we establish lower bounds relating the energy $I^\e_U(u)$ 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 $I^\e_U$.