(x,t)\in\Gamma_{t_1}^{t_2}\,.$$ Proposition 5.1 on page 77 and Proposition 5.4 on page 78 in [@BC] imply, that $\overline{g}$ is a viscosity supersolution. Further we notice, since $g_1\in W^{1,\infty}(\mathbb{R}\times[0,T))$, that for all $y$ and suitable choices of $A$ and $B$ $$g_1(x,t)\leq g(y,\gamma(y))+A|x-y|+B|t-\gamma(y)|\,.$$ By Proposition 2.11 on page 302 in [@BC] $$\overline{g}(x,t)=\inf_{y}\overline{g}_y(x,t)$$ is still a supersolution. Furthermore $\overline{w}$ satisfies by construction $\overline{g}=g_1$ on $\Lambda_{t_{1}}^{t_{2}}$ and $\overline{g}\geq g$ in $\Gamma_{t_{1}}^{t_{2}}$. Hence all assumptions of the existence result (Theorem 3.1) in [@Is] are fulfilled. Therefore there exists a viscosity solution $g$ of (\[viscosity\_problem\]) such that $g_1\leq g\leq\overline{g}$. By Example 1 in [@Is], the viscosity solution is Lipschitz continuous, i.e. $g\in W^{1,\infty}(\Gamma_{t_1}^{t_2})$. For $(x,t)\in\Gamma_{t_1}^{t_2}$ and $(y,s)\in\Lambda_{t_1}^{t_2}$ we notice that $$g_1(x,t)-\overline{g}(y,s)\leq g(x,t)-g(y,s)\leq \overline{g}(x,t)-g_1(y,s)\,.$$ Using the fact that $g_1$ is Lipschitz continuous and the construction of $\overline{g}$ we deduce from the previous line $$-\|(x,t)-(y,s)\|C_1\leq g(x,t)-g(y,s)\leq C_2\|(x,t)-(y,s)\|\,,$$ which means $g\in W^{1,\infty}(\Gamma_{t_1}^{t_2}\cup \Lambda_{t_1}^{t_2})$.

Next we are going to show, that $v=\partial_x g$ is a weak solution of (\[modified\_scl\]) in $\Gamma_{t_1}^{t_2}$ in the sense of Definition \[def\_entropy\_solution\]. Since $g$ satisfies (\[viscosity\_problem\]) almost everywhere, it follows for a $\psi\in C^{\infty}_c(\mathbb{R}\times[0,t_2))$ $$\label{int1}
\int_{\Gamma_{t_1}^{t_2}}\partial_x\psi\partial_{t}g+f(\partial_xg)\partial_{x}\psi\,dx\,dt=0\,.$$ We denote the outer unit normal vector of $\Gamma_{t_1}^{t_2}$ by $n$. Integrating (\[int1\]) twice by parts gives $$\label{int2}\begin{split}
\int_{\Gamma_{t_1}^{t_2}}\partial_x\psi\partial_{t}g\,dx\,dt&=\int_{\partial\Gamma_{t_1}^{t_2}}g\begin{pmatrix}-\partial_t\psi\\\partial_x\psi\end{pmatrix}\cdot n\,d\sigma+ 