 Affine Sobolev space

For $1\le p<\infty$, an open set $\Omega$ and $\phi \in W^{1,p}(\Omega )$, the Affine Sobolev spaceis a translate of a closed linear subspace of the first-order Sobolev space. This definition makes sense without assuming an ordinary boundary trace.

On a bounded Lipschitz domain, the Sobolev trace theorem identifies it exactly with the Sobolev space elements having the same boundary trace as $\phi$, since the trace kernel is $W_0^{1,p}(\Omega )$. It is the natural admissible set for a variational Dirichlet problem. Specifying a trace datum alone also requires an extension $\phi$ in the stated Sobolev space; arbitrary boundary data need not admit one.