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!