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



Optimal Lifting for BV Functions into $ S^1$

Abstract: Let $ \Omega \subset \mathbb{R}^N$ be an open set and $ u:\Omega \to S^1$ a measurable function. A lifting of $ u$ is a measurable function $ \varphi :\Omega \to \\ mathbb{R}$ such that $ u(x) = e^{i \varphi(x) }$ for a.e. $ x\in \Omega$. If $ u$ has some regularity one may ask whether or not $ \varphi$ 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\in BV(\Omega, S^1)$ and to determine its total variation, i.e.,

$\displaystyle E(u)=\min \bigg\{\int_{\Omega} \vert D \varphi\vert \, :\, \varphi \in BV(\Omega, \mathbb{R}),\, e^{i\varphi}=u \textrm{ a.e. in } \Omega \bigg\}.$ (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 $ \Gamma-$limit of the functionals $ \{F_\varepsilon\}_{\varepsilon \downarrow 0}$ defined as

$\displaystyle F_\varepsilon(\varphi)=\varepsilon \int_\Omega \vert\nabla \varph... ...a \vert u-e^{i\varphi}\vert^p, \quad \forall \varphi \in H^1(\Omega,\mathbb{R})$ (2)

where $ p\in (1,+\infty)$.