I have an issue in a proof I face. I have a function $f$ in $H^s_0(U)$, where $U$ is bounded with $C^1$-boundary. Then I would like to say that $f=0$ on $\partial U$ however this is not necessarily the case, no?
I think, I need make sure $f$ is also continuous and that a trace operator exists, but this seems only to be ensured if the domain’s boundary is in addition Lipschitz.
Or does this hold anyway?