Approximation of BV space-defined Functionals Containing Piecewise Integrands with L¹ Condition

Abstract

We prove an approximation result for a class of functionals $% %TCIMACRO{\TeXButton{mathcal G}{\mathcal{G}}}% %BeginExpansion \mathcal{G}% %EndExpansion (u)=\int_{\Omega }\varphi (x,Du)$ defined on $BV\left( \Omega \right) $ where $\varphi (\cdot ,Du)\in L^{1}\left( \Omega \right) ,$ $\Omega \subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{N}$ bounded, $\varphi (x,p)$ convex, radially symmetric and of the form \begin{equation*} \varphi (x,p)=\left\{ \begin{tabular}{ll} $g(x,p)$ & if $|p|\leq \beta $ \\ $\psi (x)|p|+k(x)$ & if $|p|>\beta .$% \end{tabular}% \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 $% %TCIMACRO{\TeXButton{mathcal G}{\mathcal{G}}}% %BeginExpansion \mathcal{G}% %EndExpansion (u_{k})\rightarrow %TCIMACRO{\TeXButton{mathcal G}{\mathcal{G}}}% %BeginExpansion \mathcal{G}% %EndExpansion (u).$ 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.