On $L^{∞ }$ and $L^{2}$ Bounds for Weak Solutions of Time Flowsfor Certain Functionals of Linear Growth (European Journal of Pure and Applied Mathematics, 2024)
We use recent approximation results in $BV$ space to derive $L^{\infty }$bounds for $u,$ $L^{\infty }$ bounds in time for the $BV$ seminorm $\int|Du| $ of $u,$ and $L^{2}$ bounds for $u_{t}$ for the weak solution $u\inC([0,\infty );L^{2}\left( \Omega \right) \cap BV\left( \Omega \right) ),$ $%\Omega \subset%TCIMACRO{\U{211d} }%%BeginExpansion\mathbb{R}%EndExpansion^{N}$ open and bounded, of the time flow\begin{equation*}\frac{\partial u}{\partial t}=\func{div} abla _{p}\varphi (x,Du)-\lambda(u-u_{0}),\text{ }\lambda >0,\text{ }u(0,x)=u_{0}.\end{equation*}%We assume Neumann boundary condition and $\varphi (x,p)$ is in a class oflinear growth functions in $p.$ Importantly, $\varphi (\cdot ,p)\inL^{1}\left( \Omega \right) $ in contrast to the classical results stated in\cite{ACM} where $\varphi $ has a continuity assumption in the $x$ variable.We also use the convergence of the solution above to derive an $L^{\infty }$bound for the solution $u^{\ast }$ to the corresponding stationary problem,since $u(t)\rightarrow u^{\ast }$ in $L^{1}\left( \Omega \right) .$
Keywords: bounded variation, weak solution, variational problems, linear growth