= Affine Sobolev space
{c}
{title2=$\varphi+W_0^{1,p}(\Omega)$}
For $1\leq p<\infty$, an <open set> $\Omega$ and $\varphi\in W^{1,p}(\Omega)$, the <Affine Sobolev space>
$$
\varphi+W_0^{1,p}(\Omega)=\{\varphi+w:w\in W_0^{1,p}(\Omega)\}
$$
is 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 $\varphi$, 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 $\varphi$ in the stated <Sobolev space>; arbitrary boundary data need not admit one.
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