Approximation of BV space-defined functionals containing piecewise integrands with L1 condition

Abstract

We prove an approximation result for a class of functionals $\mathcal{G}(u)={\int }_{\mathrm{\Omega }}\phi (x,Du)$ defined on $BV\left(\mathrm{\Omega }\right)$
where $\phi (\cdot ,Du)\in L^1\left(\mathrm{\Omega }\right),$ $\mathrm{\Omega }\subset {\mathbb{R}}^N$ bounded, $\phi (x,p)$ convex, radially symmetric and of the form
$$\phi (x,p)=\{\begin{array}{ll}\$g(x,p)\$& if\$|p|\le \beta \$\\ \$\psi (x)|p|+k(x)\$& if\$|p|>\beta .\$\end{array}$$%
We show for each $u\in BV\left(\mathrm{\Omega }\right)\cap L^p\left(\mathrm{\Omega }\right),$ $1\le p<\mathrm{\infty },$ there exist $u_k\in W^{1,1}\left(\mathrm{\Omega }\right)\cap C^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\cap L^p\left(\mathrm{\Omega }\right)$ so that $\mathcal{G}(u_k)\to \mathcal{G}(u).$ Approximation theorems in $BV$ are used to prove existence results for
the strong solution to the time flow $u_t=\text{func}div({\mathrm{\nabla }}_p\phi (x,Du))$ in $L^1((0,\mathrm{\infty });BV\left(\mathrm{\Omega }\right)\cap L^p\left(\mathrm{\Omega }\right)),$ typically with additional boundary
condition or penalty term in $u$ to ensure uniqueness. The functions in this
work are not covered by previous approximation theorems since for fixed $p$
we have $\phi (x,p)\in L^1\left(\mathrm{\Omega }\right)$ which do not in
general hold for assumptions on $\phi$ in earlier work.