= Boundary trace of a function
{title2=$\gamma u=u|_{\partial\Omega}$}
= Boundary trace
{synonym}
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 $f_j(t)=\partial_x^ju(0,t)$ 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.
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