Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\infty}(\Omega)$$
Suppose that there exist a set $\Gamma$ of positive measure such that $\nabla u=0, a.e.\ x\in\Gamma$. How can one show that $\tilde{u}=0,\ a.e.\ x\in\Gamma$?
Any help is appreciate. I have to solve this problem without considering the fact that $\tilde{u}=-\Delta u$. I know that i have to take some "nice" $v\in C_0^{\infty}(\Omega)$, but what $v$?
Edit 1: Why im trying to solve this problem? Let $p\geq 1$. Suppose $u\in W_0^{1,p}(\Omega)$ and $\tilde{u}\in L^q(\Omega)$ $(\frac{1}{p}+\frac{1}{q}=1)$ satisfies $$\int_\Omega|\nabla u|^{p-2} \nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\infty}(\Omega)$$
In this case we dont have enough regularity to show that $\tilde{u}=-\Delta_p u$, so i need a more direct aproach, and consequently i think that this aproach is the same for both cases.
Lastly we can have the $\Phi$-laplacian too and considering the spaces where it makes sense, we have $$\int_\Omega\Phi(|\nabla u|)\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\infty}(\Omega)$$
Edit 2: In general, consider $\sigma$ a vector valued function. Suppose $\sigma$ is in a convenient space and $$\int_\Omega \sigma\cdot\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\infty}(\Omega)$$
where, $\cdot$ stands for inner product. Now i want to conclude the same thing for all the cases above.
Note: $\tilde{u}$ is called weak divergence of $\sigma$.
Edit 3: In any open set $U$ contained in $\Gamma$, we can take functions with compact supoort in $U$ and conclude that $\tilde{u}=0,\ a.e.\ x\in U$, but in the general case, how to proceed? Maybe this is not valid in a general set with positive measure? Any opinion?
Thanks
