Morning!
In pde often there are boundary conditions for Sobolev functions. For example, "If $f\leq g$ on $\partial E$, then..." where $E$ is a domain in $\mathbb{R}^n$ and $f,g\in W^{1,p}(E)$ for some $1\leq p<\infty$. This condition basically assumes that $f,g$ both have a trace on $\partial E$, as they are only defined on E and otherwise this condition can't be fulfilled - or do I misunderstand this and it doesn't assume anything regarding existence of traces but is just a condition? Meaning that, if they have a trace then ... holds and otherwise (if they have no trace for example) ... simply isn't valid?
Functions in Sobolev spaces have a well defined trace, which belongs to $L^p(\partial E)$. You have to assume that the boundary is somehow regular, say Lipschitz.
Check the article “Trace Theorem” on Wikipedia.