Approximation of BV space-defined functionals containing piecewise integrands with $L^{1}$ condition

Abstract

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 $u\in BV\left( \Omega \right) \cap L^{p}\left( \Omega
\right) ,$ $1\leq p<\infty ,$ there exist $u_{k}\in W^{1,1}\left( \Omega
\right) \cap C^{\infty }\left( \Omega \right) \cap L^{p}\left( \Omega
\right) $ so that $
\mathcal{G}%
%EndExpansion
(u_{k})\rightarrow
%TCIMACRO{\TeXButton{mathcal G}{\mathcal{G}}}%
%BeginExpansion
\mathcal{G}%
%EndExpansion
(u).$ Approximation theorems in $BV$ are used to prove existence results for
the strong solution to the time flow $u_{t}=\func{div}\left( \nabla
_{p}\varphi (x,Du\right) )$ in $L^{1}((0,\infty );BV\left( \Omega \right)
\cap L^{p}\left( \Omega \right) ),$ 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 $\varphi (x,p)\in L^{1}\left( \Omega \right) $ which do not in
general hold for assumptions on $\varphi $ in earlier work.