We define a mapping which with each function $u\in L^{\mathrm{\infty }}\left(0,T\right)$ and an admissible value of $r>0$ associates the function $\xi$ with a prescribed initial condition ${\xi }^0$ which minimizes the total variation in the $r$-neighborhood of $u$ in each subinterval $\left[0,t\right]$ of $\left[0,T\right]$. We show that this mapping is non-expansive with respect to $u$, $r$ and ${\xi }^0$, and coincides with the so-called play operator if $u$ is a regulated function.

We consider the hyperspace $CL\left(X\right)$ of nonempty closed subsets of completely metrizable space $X$ endowed with the Wijsman topologies ${\tau }_{W_d}$. If $X$ is separable and $d$, $e$ are two metrics generating the topology of $X$, every countable set closed in $\left(CL\left(X\right),{\tau }_{W_e}\right)$ has isolated points in $\left(CL\left(X\right),{\tau }_{W_d}\right)$. For $d=e$ , this implies a theorem of Costantini on topological completeness of $\left(CL\left(X\right),{\tau }_{W_d}\right)$. We show that for nonseparable $X$ the hyperspace $\left(CL\left(X\right),{\tau }_{W_d}\right)$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.

We state and prove a chain rule formula for the composition $T\left(u\right)$ of a vector-valued function $u\in W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ by a globally Lipschitz-continuous, piecewise $C^1$ function $T$. We also prove that the map $u\to T\left(u\right)$ is continuous from $W^{1,r}\left(\mathrm{\Omega };{\mathbb{R}}^M\right)$ into $W^{1,r}\left(\mathrm{\Omega }\right)$ for the strong topologies of these spaces.

Given an open, bounded and connected set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Lipschitz boundary and volume $\left|\mathrm{\Omega }\right|$, we prove that the sequence ${\mathcal{F}}_k$ of Dirichlet functionals defined on $H^1\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$, with volume constraints $v^k$ on $m\ge 2$ fixed level-sets, and such that ${\sum }_{i=1}^m{v}_i^k<\left|\mathrm{\Omega }\right|$ for all $k$, $\mathrm{\Gamma }$-converges, as $v^k\to v$ with ${\sum }_{i=1}^m{v}_i^k=\left|\mathrm{\Omega }\right|$, to the squared total variation on $BV\left(V;{\mathbb{R}}^d\right)$, with $v$ as volume constraint on the same level-sets.