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.