Some related earlier discussion can be found here.

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary, $\mathcal H^{N-1}(\partial\Omega)<\infty$ and $u\in SBV(\Omega)$. Then $$ Du = \nabla u\mathcal L^N\lfloor\Omega + (u^+-u^-)\nu_u\mathcal H^{N-1}\lfloor S_u, $$ where $S_u$ is the jump set of $u$, $\nu_u$ is a measurable field of normals on $S_u$, and $u^+$ and $u^-$ are the corresponding one-sided approximate limits. In addition, suppose the following three properties: $$\|T(u)\|_{L^\infty(\partial \Omega)}< C_1\tag 1$$ $$ \|\nabla u\|_{L^\infty(\Omega)}<C_2 \tag 2 $$ $$ \int_{S_u}|u^+-u^-|\,d\mathcal H^{N-1}<C_3 \tag 3 $$ Here $C_1$, $C_2$, and $C_3$ are positive constants and $T$ denotes the trace operator.

Question:Can I estimate $\|u\|_{L^\infty}$ by the constants $C_1$, $C_2$, $C_3$ and constants that depend on $N$ and $\Omega$? If yes, can I further weaken (2) to $$ \int_\Omega|\nabla u|^2<C_2 ? $$

Thank you!