Boundary trace of a function ($\gamma u=u{|}_{\partial \Omega }$)

For a sufficiently regular function on a domain $\Omega$, its boundary trace is the restriction to $\partial \Omega$, using the limit from within the domain when necessary. In a half-line evolution problem, the traces $f_j(t)={\partial }_x^{j}u(0,t)$ are boundary values of the spatial derivatives. An initial value does not generally determine every such trace without the partial differential equation and suitable boundary conditions. The Sobolev trace operator extends classical restriction continuously to appropriate Sobolev spaces, where pointwise values need not be defined.