These days I am learning Dirichlet problems, which usually view the boundary condition $\partial_x^\beta \phi|_{\partial\Omega}=0$ to the condition that the solution lives in $H^m_0$.
This is acceptable if the boundary is smoot enough, as we can apply trace theorem to show that if $u \in H^m$, $u\in H^m_0 \iff u|_{\partial \Omega}=0$.
but is it true when the boudary is not smooth enough?
As usual notion, here $\Omega$ denotes a bounded open domain with $C^1$ boundary.
$H^m=\{u\in L^2(\Omega)|\partial_x^\beta u\in L^2(\Omega)\ \forall |\beta|\leq m \}$.
$H^m_0$ is the closure of $C_0^\infty(\Omega)$, which is all the smooth functions with compact support in $\Omega$
The question is, suppose $\phi\in C^m(\Omega)$, then is $\phi\in H^m_0 \iff \partial_x^\beta\phi|_{\partial \Omega}=0\ \forall |\beta|\le m-1$ true or not? Where the restriction is taken in classical sence.