Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write 
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu\_u\mathcal H^{N-1}\lfloor S\_u
$$
where $S\_u$ is the jump set of $u$.


Then define set $\mathcal F$ such that 
$$
\mathcal F:=\{v\in SBV(\Omega)\cap L^\infty(\Omega),\|\nabla v\|\_{L^\infty}\leq C, T[v]=T[u],S\_v\subset S\_u\}
$$
where $C>0$ is a constant.


Then, do I have that $v\in\mathcal F$ is $L^\infty$ uniformly bounded?


If not, how may I modify $\mathcal F$ so that my requirement is true? Maybe require more on jump set? Since I feel my conjecture may not be true since I give no requirement on how much it can jump for a function $v\in\mathcal F$.


Thank you!



Is this set of function belongs to $L^\infty$?
Let $\Omega=[-1,1]$ and $u(x)=u(-x)=0$ for $x\in[2^{-2k+1},2^{-2k+2})$ and $u(x)=u(-x)=2^{-k}$ for $x\in[2^{-2k},2^{-2k+1})$, $k=1,\ldots,n,\ldots$.


Doesn't this $u$ belong to your space $SBV\cap L^\infty$ ? Because if no condition is imposed on the jumps, the set $\mathcal F$ clearly contains piecewise constant functions with the same jump set $S\_u=\{\pm 2^{-n}:\ n=1,\ldots\}$ and arbitrarily large values.


