Abstract

We consider the generalized elastica functional defined on L1(R2)\lp{1}(\R^2) as F(u)={∫R2∣∇u∣(α+β∣div⁡∇u∣∇u∣∣p) dx,if u∈C2(R2),+∞else,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>1p>1, α>0\alpha>0, β≥0\beta\geq 0. We study the L1\lp{1}-lower semicontinuous envelope F‾\oF of FF and we prove that, for any u∈BV(R2)u\in\BV(\R^2), F‾(u)\oF(u) can be represented by a coarea-type formula involving suitable collections of W2,p\wpq{2}{p} curves that cover the essential boundaries of the level sets {x, u(x)>t}\{x,\,u(x)> t\}, t∈Rt\in\R

Author information

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Simon Masnou

Institut Camille Jordan, Université de Lyon 1, 43 bd du 11 novembre 1918, 69622 Villeurbanne-Cedex, France
and: CNRS UMR 5208, Villeurbanne-Cedex, France

masnou@math.univ-lyon1.fr

Giacomo Nardi

Laboratoire Jacques-Louis Lions, Université P. et M. Curie Paris 6, F-75005 Paris, France
and: CNRS UMR 7598, Paris, France

nardi@ann.jussieu.fr

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Suggested citation

S. Masnou, G. Nardi. “A Coarea-Type Formula for the Relaxation of a Generalized Elastica Functional.” Journal of Convex Analysis 20 (2013), No. 3, 617–653. https://doi.org/10.68381/jca20036

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press.