Let $u_i \in C^1(\Omega)$ with $|\nabla u_i|>0$ in a simply connected region $\Omega$, and $u_1=u_2$ on $\partial \Omega$. Assume $$ \nabla u_i(x) \cdot V_i (x)=|\nabla u_i(x)||V_i(x)|, \ \ \forall x \in \Omega$$ for two vector fields $V_i\in L^{\infty}(\Omega)$, $i=1,2$. I wonder if $||V_2-V_1||_{L^2}$ being small would imply $u_2-u_1$ is also small in some norm (ideally in $H^1_0$ norm).