Boundary-constant-preserving smooth approximation

Write $O_{-}$ and $O_{+}$ for the open neighborhoods where $u$ has the two prescribed constant values. The set $V=O_{+}\setminus B_{-}$ is open and contains the compact set $B_{+}$. By [F1], take a smooth ${\chi }_{+}:R\to [0,1]$ supported in $V$ and equal to one on an open neighborhood $U_{+}$ of $B_{+}$, shrinking $U_{+}$ inside $O_{+}$. Put $U_{-}=O_{-}\setminus \mathrm{supp}{\chi }_{+}$; this is an open neighborhood of $B_{-}$. Set $h=c_{-}+(c_{+}-c_{-}){\chi }_{+}$. Then $h$ is smooth up to the boundary, equals $c_{-}$ on $U_{-}$ and $c_{+}$ on $U_{+}$, and $v=u-h$ vanishes on $U_{-}\cup U_{+}$.

The smooth dual metric and compactness give a finite $M\ge 1$ with $\mathrm{\mid }dh{\mathrm{\mid }}_k\le M$ on $R$; for empty $R$ take $M=1$. For any piecewise $C^1$ curve $\gamma$ joining $p$ to $q$, refine into finitely many chart pieces. The half-space chain rule, using smooth local extensions also when a piece touches the boundary, gives $\mathrm{\mid }(h\circ \gamma {)}^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }dh(\dot{\gamma })\mathrm{\mid }\le M\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_k$. The scalar fundamental theorem on each piece and the triangle inequality therefore give $\mathrm{\mid }h(p)-h(q)\mathrm{\mid }\le M{L}_k(\gamma )$. Taking the infimum over curves gives $\mathrm{\mid }h(p)-h(q)\mathrm{\mid }\le M{d}_k(p,q)$ on each component. Distinct components impose no finite-distance constraint. Thus $h$ is Lipschitz under the convention in the Statement, and if $u$ has Lipschitz bound $L$, then $v=u-h$ has bound $L+M$. Its support is closed in compact $R$ and disjoint from the open neighborhood $U_{-}\cup U_{+}$ of all of $\mathrm{\partial }R$, hence is compactly contained in $R^{\circ }$.

Apply [F2], under [A1], to the Lipschitz $v$ from step 2.1, obtaining smooth $v_j$ supported in a single compact $K\subset R^{\circ }$, with uniform error tending to zero and ${\int }_R(\mathrm{\mid }{v}_j-v\mathrm{\mid }+\mathrm{\mid }d{v}_j-dv{\mathrm{\mid }}_k)d{V}_k\to 0$. Define $u_j=h+v_j$. On the fixed open neighborhoods $U_{-}\setminus K$ and $U_{+}\setminus K$ of the respective boundary parts, every $u_j$ has the exact prescribed constant. Weak differentiation is linear: the coordinate test identities add, so $d{u}_j-du=d{v}_j-dv$ almost everywhere, while $u_j-u=v_j-v$ pointwise. Thus the uniform errors and the displayed Sobolev errors are exactly those of $v_j-v$ and tend to zero. The sum is smooth up to the boundary because $h$ is, and $v_j$ vanishes near it. The conventions of [F4] include dimension zero and the empty case, completing all assertions.