Trace of $H^1(\Omega)$ function onto regular hypersurface inside $\Omega$ (and not the boundary)?

Question

Let $\Omega$ be a smooth domain in 2D and let $S$ be a closed smooth surve inside $\Omega$

Do functions in $H^1(\Omega)$ have a trace on $S$ and does it follow that $u \in H^1(\Omega) \implies u|_{S} \in H^{\frac 12}(S)$?

I.e. is the fractional Sobolev space the correct trace space?