Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ Let us denote by $W_0^{1,2}$ the closure in $W^{1,2}$ of infinitely smooth functions with compact support in $\Omega$.

Let $u\in W^{1,2}_0\cap C(\bar \Omega)$. Is it true that $u$ vanishes on $\partial \Omega$?

Apologies if this question is not of the research level.

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ Let us denote by $W_0^{1,2}$ the completion of the space of smooth compactly supported functions in $\Omega$ with respect to this norm.

Let $u\in W^{1,2}_0\cap C(\bar \Omega)$. Is it true that $u$ vanishes on $\partial \Omega$?

Apologies if this question is not of the research level.