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

We consider the generalized elastica functional defined on

as

F(u)=\left\{\begin{array}{ll} \displaystyle\int_{\mathbb{R}^2}|\nabla u|(\alpha+\beta|\operatorname{div}\frac{\nabla u}{|\nabla u|}|^p)\,dx,&\text{if $u\in {{\mathrm{C}}}^2(\mathbb{R}^2),$}\\[6mm] +\infty&\text{else},\end{array}\right. F ( u ) = { ∫ R 2 ∣ ∇ u ∣ ( α + β ∣ div ⁡ ∇ u ∣ ∇ u ∣ ∣ p )   d x , if  u ∈ C 2 ( R 2 ) , + ∞ else ,

where

p>1 p > 1

,

\alpha>0 α > 0

,

\beta\geq 0 β ≥ 0

. We study the

{\mathrm L}^{1} L 1

-lower semicontinuous envelope

\overline{F} F ‾

of

F F

and we prove that, for any

u\in{{\mathrm{BV}}}(\mathbb{R}^2) u ∈ B V ( R 2 )

,

\overline{F}(u) F ‾ ( u )

can be represented by a coarea-type formula involving suitable collections of

{\mathrm W}^{2,p} W 2 , p

curves that cover the essential boundaries of the level sets

\{x,\,u(x)> t\} { x ,   u ( x ) > t }

,

t\in{\mathbb{R}} t ∈ R