Optimal lifting for BV(S1,S1)

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g \in BV(S^1,S^1)$\end{document}, we solve the following variational problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(g)=\inf \left\{ \int_{S^1} |\dot{\varphi}| \, :\, \varphi \in BV(S^1, \mathbb{R}), e^{i\varphi}=g \textrm{a.e. on} S^1\right\}$$\end{document} and we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E(g)\leq 2\vert g\vert _{BV}$\end{document}.