Let $\Omega=[-L,L] \subset \mathbb{R}$, and let $n=\dfrac{u_x}{|u_x|}$. Now my question is what are the conditions on $\gamma(n)$ and $u_x$ so that we have
$$\gamma^2(n) u_x \in L^1$$
i.e. $\gamma^2(n) u_x$ integrable.
where $\gamma$ is any function as general as we can take it.
If we assume $u_x\in L^1(\Omega)$, then if we assertain the assumption that $|\gamma(\pm 1)|\leq K$ for some $K$ finite, then:
$\int_\Omega|\gamma^2(n)u_x|dx=\int_{-L}^{L}|\gamma^2(\pm1)||u_x|dx\leq K^2\int_{-L}^{L}|u_x|dx\lt\infty$, thus then $\gamma^2(n)u_x\in L^1(\Omega)$
Equivalently, since $\Omega=[-L,L]$ where $L$ is finite, then $u_x\in L^1$ if it is bounded.