International audienceWe characterize the couples $(s,p)$ with the following property: if $u$ is a complex-valued unimodular map in $W^{s,p}$, then $u$ has (locally) a phase in $W^{s,p}$

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Let Ω ⊂ R 2 be a simply connected domain, let ω be a simply connected subdomain of Ω, and set A = Ω \ ω. Suppose that J is the class of complex-valued maps on the annular domain A with degree 1 both on ∂Ω and on ∂ω. We consider the variational problem for the Ginzburg-Landau energy E λ among all maps in J . Because only the degree of the map is prescribed on the boundary, the set J is not necessarily closed under a weak H 1 -convergence. We show that the attainability of the minimum of E λ over J is determined by the value of cap(A)-the H 1 -capacity of the domain A. In contrast, it is known, that the existence of minimizers of E λ among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap(A) π (A is either subcritical or critical), we show that the global minimizers of E λ exist for each λ > 0 and they are vortexless when λ is large. Assuming that λ → ∞, we demonstrate that the minimizers of E λ converge in H 1 (A) to an S 1 -valued harmonic map which we explicitly identify. When cap(A) < π (A is supercritical), we prove that either (i) there is a critical value λ 0 such that the global minimizers exist when λ < λ 0 and they do not exist when λ > λ 0 , or (ii) the global minimizers exist for each λ > 0. We conjecture that the second case never occurs. Further, for large λ, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices-a vortex of degree 1 near ∂Ω and a vortex of degree −1 near ∂ω.

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Let g : I = (0, 1) → S 1 . If g ∈ VMO, we may write g = e iϕ for some ϕ ∈ VMO; this ϕ is unique modulo 2π (see [13] and the earlier work [14]). There is no control of |ϕ| BMO in terms of |g| BMO , since we always have |g| BMO ≤ 2 and |ϕ| BMO can be arbitrarily large; recall, however, that, when |g| BMO is sufficiently small, there is a linear estimate |ϕ| BMO ≤ C|g| BMO (see [13, theorem 4], [14], and Remark 0.2 below).We are going to establish that a norm slightly stronger than |g| BMO does control |ϕ| BMO . Consider, for 1 < p < ∞, 0 < s < 1, the fractional Sobolev space W s, p (I ), equipped with its standard seminorm |g| s, p =