Trace operator

Summary

The trace operator gives boundary values to functions in a Sobolev space, which in general are only defined up to sets of measure zero. Let $U$ be a bounded open set with $C^1$ boundary, and let $1 \le p < \infty$. The trace operator extends the restriction $u \mapsto u|_{\partial U}$ to all of $W^{1,p}(U)$, as a bounded linear map into $L^p(\partial U)$. It gives a precise meaning to boundary conditions such as $u = g$ on $\partial U$ for weak solutions of boundary value problems.