Let $\Omega$ be an open bounded set of $\mathbb{R}^N$ and $u\in H_0^1(\Omega)$. Suppose $|\nabla u|>1$ on a set of positive measure, then by inner regularity of Lebesgue measure, there exists a compact set with positive measure $$K\subset \{|\nabla u| > 1\}.$$
Now if I define another function $v$ such that
$\nabla u = \nabla v$ on $K$, and
$\nabla v = 0$ on $K^c$.
Is $v\in H_0^1(\Omega)$?
If not, is there any other way to define this function $v$? I want the inner product $(u,v)_{H_0^1} = \int_\Omega \nabla u\cdot \nabla v$ to capture (part of) the change when $|\nabla u| > 1$ and ignore all the changes when $|\nabla u| \leq 1$.
Thank you very much!