Abstract. We study properties of Ginzburg-Landau functionals I U (·), defined for functions u ∈ W 1,n (U ; R n ), where U ⊂ R n . 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 .

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Abstract. We study the semiclassical limit of subcritical focussing NLS with a potential, for initial data of the form s(, where s is the ground state of an associated unscaled problem. We show that in the semiclassical limit, the solution has roughly the form s(, and we show that the approximate center of mass x (·) converges to a solution of the equation x = −DV (x), x(0) = x 0 , x (0) = v 0 as → 0.

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems -under certain sharp conditions -for minimizers of the general least gradient problemand ϕ(x, ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular we prove that if a ∈ C 1,1 (Ω) is bounded away from zero, then minimizers of the weighted least gradient problem inf u∈BV f Ω a|Du| are unique in BV f (Ω). We construct counterexamples to show that the regularity assumption a ∈ C 1,1 is sharp, in the sense that it can not be replaced by a ∈ C 1,α (Ω) with any α < 1.