Let $\Omega\subset \mathbb{R}^2$ be an open and bounded set. Set $M=\{f\in L^2(\Omega):\nabla f\in L^1(\Omega)^2\}$. Define the functional (generalized ROF model) as:
$$ J(f)=\begin{cases} \displaystyle \int_\Omega |f_0-Rf|^2+\alpha\int_\Omega \phi(|\nabla f|), \quad f\in M \\[0.2cm] +\infty, \quad f\in BV(\Omega)\setminus M\end{cases}. $$ Here $\phi:\mathbb{R}\to\mathbb{R}$ is a strictly convex, non-decreasing function with $\phi(0)=0$ and satisfies a linear growth property: $\exists a>0, b\ge 0$ such that $ax-b\le\phi(x)\le ax+b$, $f_0$ is the initial noisy image.
The operator $R:L^2(\Omega)\to L^2(\Omega)$ is linear and continuous.
How to show that the functional $J$ is not lower semi-continuous in $BV-w^*$ topology ? By $BV-w^*$ topology we mean that $u_k\to u$ strongly in $L^1(\Omega)$ and $Du_k\rightharpoonup Du$ weakly in the sense of Radon measures, i.e. $\int_\Omega \varphi Du_k\to\int_\Omega\varphi Du$ for all $\varphi\in C_0(\Omega)^2$.