Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where $\Sigma$ is an embedded hypersurface. Also, if $X$ denotes the unit normal vector field to the hypersurface $\Sigma$, we additionally have $Xu = 0$. I am trying to find a reference that says that $u$ is locally Lipschitz. I am looking in Gilbarg-Trudinger, but am unable to find it. Thanks for your help!

Note: I edited the question after Denis Serre's answer (see comments below).

Further note: I edited the question again after Michael Renardy and Connor Mooney's comments.