A coarea-type formula for the relaxation of a generalized elastica functional (Q2850725)

Let \(\overline{F}(u),u\in L^1(\mathbb R^2)\) be the relaxation of the generalized elastica functional NEWLINE\[NEWLINE F(u)= \begin{cases} \int_{R^2}|\nabla u|\Big (\alpha+\beta\Big |\text{div} \frac{\nabla u}{|\nabla u|}\Big |^p \Big ) dx,\; p>1,\;\alpha>0,\; \beta\;\geq 0,\\ \infty, \text{otherwise}, \end{cases} NEWLINE\]NEWLINE i.e. NEWLINE\[NEWLINE \overline{F}(u)=\inf\Big \{\lim_{h\rightarrow \infty}\inf F(u_h):\{u_h\}\subset C^2(\mathbb R^2),u_h\rightarrow u\; \text{in the sense of}\; L^1(\mathbb R^2) \Big\}. NEWLINE\]NEWLINE In the work, the \(L^1(R^2)\)-lower semicontinuity of the functional \(\overline{F}(u)\) is proved. Moreover, for any \(u\in BV(\mathbb R^2)\) the functional \(\overline{F}(u)\) is represented by a coarea-type formula. Finally, examples are discussed.