Radu Ignat Laboratoire Jacques-Louis Lions Université Paris ignat@clipper.ens.fr

Optimal Lifting for BV Functions into S^{1}

Abstract: Let Ω ⊂ ℝ^{N} be an open set and u : Ω → S^{1} a measurable function. A lifting of u is a measurable function φ : Ω → mathbbR such that u(x) = e^{iφ(x)} for a.e. x ∈ Ω. If u has some regularity one may ask whether or not φ can be chosen with some regularity as well. Motivated by the study of the Ginzburg-Landau equation and the degree theory, there has been recently much research on this topic for certain spaces of functions with values into the circle S^{1} (such as Sobolev spaces, BMO, VMO, BV).

The aim of this talk is to construct an optimal lifting for u ∈ BV (Ω, S^{1}) and to determine its total variation, i.e.,

E(u) = min { ∫_{Ω} |Dφ| : φ ∈ BV (Ω, ℝ), e^{iφ} = u a.e. in Ω}.  (1)

The formula of E(u) will involve the notion of minimal connection between surface singularities of dimension N − 2 of u. We also discuss the relation between optimal BV lifting and minimizers of the Γ−limit of the functionals {F_{ε}}_{ε↓0} defined as

F_{ε}(φ) = ε ∫_{Ω} |∇φ|^{2} + \frac{1}{ε} ∫_{Ω} |u − e^{iφ}|^{p}, ∀φ ∈ H^{1}(Ω, ℝ)  (2)

where p ∈ (1, +∞).