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

\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 $\{x,\,u(x)> t\}$, $t\in\R$