I'm trying to make some general statement about the operator mapping of the Eikonal equation (not an expert on PDEs nor functional analysis), namely
\begin{eqnarray}|\nabla u(x)| = f(x)\end{eqnarray}
for $f(x) \in C^0(\Omega;\mathbb{R}^d)$ where $\Omega$ is a bounded domain. I believe we can assume since $f(x) \in C^0$ that $u(x)$ has a viscosity solution that is unique and exists approaching the solution $u(x) \in C^0$ from the papers [1], [2] below. But, now I want to consider the continuity of the operator namely:
\begin{eqnarray}\mathcal{G}: C(\Omega) \rightarrow C(\Omega)\end{eqnarray}
where the operator is defined as the mapping from $f(x)$ into the solution $u(x)$. From an intuitive standpoint, is it ever possible that this mapping is continuous given $f_1, f_2 \in C(\Omega)$ are such that for $U>0$, $\|f_1\|_\infty, \|f_2\|_\infty < U$ even if $\nabla u(x)$ is not necessarily defined everywhere?
I have tried proving continuity using approaches such as Poincaré on $L_2$ as something like \begin{eqnarray} \|G(f_1) - G(f_2)\|_{L_2} &\leq& c \|\nabla (G(f_1)-G(f_2))\|_{L_2} \\ &\leq& c\|\nabla G(f_1) - \nabla G(f_2)\|_{L_2} \end{eqnarray} But then, applying any sort of norm forces $\| \nabla G(f_1) - \nabla G(f_2) \|_{L_2}$ to become a + removing any sort of Lipschitz or Hölder continuity leaving me to beleive it is not so likely. Any sort of ideas about whether the intuition that it is actually possible to prove continuity would be extremely helpful.
[1] H. Ishii. A simple, direct proof of uniqueness for solutions of the hamilton-jacobi equations of eikonal type. Proceedings of The American Mathematical Society - PROC AMER MATH SOC, 100:247–247, 02 1987. URL: https://www.ams.org/journals/proc/1987-100-02/S0002-9939-1987-0884461-3/S0002-9939-1987-0884461-3.pdf
[2] S. N. Kruˇzkov. Generalized solutions of the hamilton-jacobi equations of eikonal type. i. formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions. Mathematics of the USSR-Sbornik, 27(3):406, apr 1975. URL: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3720&option_lang=eng
If you are still interested, you can get continuity from the maximum principle (or rather, comparison principle), which says that $u_1\leq u_2$ whenever $f_1\leq f_2$ (assuming here that $u_1=u_2=0$ on the boundary).
To see how to do this, suppose $0 < \theta \leq f_1,f_2 \leq \theta^{-1}.$ Uniform positivity is required even for uniqueness, and the upper bound is used later. Set $g_1 = \lambda f_1$ where $\lambda>0$ is chosen large enough so that $g_1 \geq f_2$, so $\lambda = \max_\Omega (f_2/f_1)$. Then the solution corresponding to $g_1$ is $\lambda u_1$, and by comparison $\lambda u_1 \geq u_2$. Therefore $$u_2 - u_1 \leq (\lambda - 1)u_1 \leq \left(\max_\Omega \frac{f_2 - f_1}{f_1}\right)u_1 \leq \theta^{-1}\|f_1-f_2\|_{L^\infty(\Omega)}u_1.$$ We can easily bound $u_1 \leq C_\Omega\theta^{-1}$, where $C_\Omega$ is something like the diameter of $\Omega$, so we get $$u_2 - u_1 \leq C_\Omega \theta^{-2}\|f_1-f_2\|_{L^\infty(\Omega)}.$$ The same inequality clearly holds for $u_1-u_2$ by swapping $u_1$ and $u_2$, so $$\|u_1 - u_2\|_{L^\infty(\Omega)} \leq C_\Omega \theta^{-2}\|f_1-f_2\|_{L^\infty(\Omega)}.$$