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Journal of Convex Analysis 20 (2013), No. 3, 617--653
Copyright Heldermann Verlag 2013

A Coarea-Type Formula for the Relaxation of a Generalized Elastica Functional

Simon Masnou
Institut Camille Jordan, Université de Lyon 1, 43 bd du 11 novembre 1918, 69622 Villeurbanne-Cedex, France

Giacomo Nardi
Laboratoire Jacques-Louis Lions, Université P. et M. Curie Paris 6, F-75005 Paris, France


\def\wpq#1#2{{\mathrm W}^{#1,#2}} \def\oF{\overline{F}} \def\cont{{\mathrm{C}}} \def\mdiv{\operatorname{div}} \def\BV{{\mathrm{BV}}} \def\lp#1{{\mathrm L}^{#1}} \def\R{\mathbb{R}} We consider the {\em generalized elastica functional} defined on $\lp{1}(\R^2)$ as $$ F(u)=\left\{\begin{array}{ll} \displaystyle\int_{\R^2}|\nabla u|(\alpha+\beta|\mdiv \frac{\nabla u}{|\nabla u|}|^p)\,dx,&\text{if $u\in \cont^2(\R^2),$}\\[6mm] +\infty&\text{else},\end{array}\right. $$ where $p>1$, $\alpha>0$, $\beta\geq 0$. We study the $\lp{1}$-lower semicontinuous envelope $\oF$ of $F$ and we prove that, for any $u\in\BV(\R^2)$, $\oF(u)$ can be represented by a coarea-type formula involving suitable collections of $\wpq{2}{p}$ curves that cover the essential boundaries of the level sets $\{x,\,u(x)> t\}$, $t\in\R$.

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