Theorem 4.4.18.C07BA2371FA3 Let $F$ be relatively closed in a Euclidean Greenian $\Omega $, and let $u\ge 0$ be superharmonic. On $\partial (\Omega \setminus F)$ prescribe $u$ on the interior boundary and zero on the original compactified boundary. Call this nonnegative lsc datum $u_0$. Its Perron solution is finite harmonic on $\Omega \setminus F$, and
\[ \widehat R_ Fu=H_{u_0}\quad \hbox{on }\Omega \setminus F. \]
On $F$ it equals $u$ except possibly at irregular interior boundary points, a polar set. For relatively closed $F$, sweeping is additive in $u$.
Proof.
The integral solution is bounded above by $u$, which is an upper competitor, and so is finite almost everywhere in each component. Positive harmonic alternatives make it finite everywhere there. Paste $H_{u_0}$ off $F$ to $u$ on $F$ and regularize. The ball comparison argument of the relatively compact sweeping theorem applies: restriction consistency compares on the intersection of each interior ball with $\Omega \setminus F$, and regular boundary limits follow by increasing continuous minorants of $u_0$. The exceptional set is precisely among the irregular interior boundary points, independently of $u$. The regularized paste is therefore superharmonic and majorizes $u$ on $F$ quasi-everywhere. Every defining majorant for $R_ Fu$ is an upper Perron competitor off $F$; the two inequalities identify the reduction. Linearity of nonnegative boundary integrals gives additivity off $F$; on $F$ both sides equal the original sum quasi-everywhere. Canonical ball averages remove this polar exception from the identity everywhere.
$\square$
Pome
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