Approximation of BV space-defined functionals containing piecewise integrands with $L^{1}$ condition (European Journal of Pure and Applied Mathematics, 2023)
We prove an approximation result for a class of functionals $\mathcal{G}%%EndExpansion(u)=\int_{\Omega }\varphi (x,Du)$ defined on $BV\left( \Omega \right) $where $\varphi (\cdot ,Du)\in L^{1}\left( \Omega \right) ,$ $\Omega \subset \mathbb{R}^{N}$ bounded, $\varphi (x,p)$ convex, radially symmetric and of the form \begin{equation*}\varphi (x,p)=\left\{ \begin{array}{ll}$g(x,p)$ & if $|p|\leq \beta $ \\ $\psi (x)|p|+k(x)$ & if $|p|>\beta .$%\end{array}%\right.\end{equation*}%We show for each $u\in BV\left( \Omega \right) \cap L^{p}\left( \Omega\right) ,$ $1\leq p
Keywords: bounded variation; conjugate function; Carathéodory function; variational problems