# Question on Sobolev spaces in domains with boundary

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.