= Logarithmic cutoff proof of harmonic uniqueness with finitely many boundary exceptions
{title2=$\int|\nabla\eta_\epsilon|^2=O(|\log\epsilon|^{-1})$}
Let $w$ be a bounded <harmonic function> in a bounded connected smooth planar region, with zero continuous boundary limits except at finitely many points. Then $w=0$. This extends <zero-boundary harmonic uniqueness> to bounded solutions with isolated boundary exceptions, without assuming their initial <Dirichlet energy> is finite.
Choose $\eta_\epsilon=0$ within distance $\epsilon^2$ of each exceptional point, $\eta_\epsilon=1$ beyond distance $\epsilon$, and interpolate linearly in the logarithm of distance. Each annular transition has $\int|\nabla\eta_\epsilon|^2=O(1/|\log\epsilon|)$. Apply the <divergence theorem> to the equation multiplied by $\eta_\epsilon^2w$. The ordinary boundary terms vanish, and the <Cauchy-Schwarz inequality> gives
$$
\int\eta_\epsilon^2|\nabla w|^2\leq4\|w\|_\infty^2\int|\nabla\eta_\epsilon|^2\longrightarrow0.
$$
Every fixed compact subset eventually has $\eta_\epsilon=1$, so the <gradient> vanishes everywhere. The nonexceptional boundary data then fix the constant to zero. Differences of bounded <harmonic extensions> with identical piecewise continuous data satisfy this hypothesis. The individual extensions can have infinite <Dirichlet energy> at a jump; the cutoff argument avoids applying an unjustified finite-energy identity to them.
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