Appllication of Stoke's Theorem in Functional Analysis

Question

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with Lipschitz boundary. Let $u,v$ be function defined on $H^1(\Omega)$ such that $u=v$ on boundary of $\Omega(\partial \Omega)$. Then can we say this: $$|u-v|_{H^1(\Omega)}=0$$ My idea is like this: $$|u-v|_{H^1(\Omega)}^2=\int_{\Omega}(\triangledown (u-v))^2dx=\int_{\partial \Omega}(u-v)^2ds=0$$ where the 2nd last equality holds from Stoke's Theorem.Any suggestions are welcome.

Answer 1

No, Let $u=0$ and $v$ be any smooth nonzero function compactly supported in $\Omega$. $|u-v|_{H^1} ≥ |v|_{L^2} > 0 $.