Is it true that if $u:\Omega\to\mathbb{R}$ is a continuous function in $W^{1,2}(\Omega)=H^1(\Omega)$ and $u(x_0)\geq u(x)$, almost for all $x\in \Omega$ then:
$\nabla u(x_0)=0$?
Here $\Omega$ is an open set in $\mathbb{R}^N$ and $x_0\in\Omega$.
That's kind of a generalization for Fermat Theorem for weak derivatives. I didn't find something similar in any book/course regarding Sobolev spaces.
I need this type of result for proving some maximum principles for parabolic pde's.
We know that: $\dfrac{\partial u}{\partial x_i}=v$ if $$-\int_{\Omega}u(x)\dfrac{\partial\phi(x)}{\partial x_i} dx=\int_{\Omega} v(x)\phi(x) dx,\ \forall\ \phi\in C^{1}_{c}(\Omega)$$.
So the question is how can we prove or disprove that $v(x_0)=0$?
I wonder if there is a theory that says something about the extreme points of a sobolev continuous function.