Logarithmic cutoff proof of harmonic uniqueness with finitely many boundary exceptions ($\int |\nabla {\eta }_{ϵ}{|}^2=O(|\log ϵ{|}^{-1})$)

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 ${\eta }_{ϵ}=0$ within distance ${ϵ}^2$ of each exceptional point, ${\eta }_{ϵ}=1$ beyond distance $ϵ$, and interpolate linearly in the logarithm of distance. Each annular transition has $\int |\nabla {\eta }_{ϵ}{|}^2=O(1/|\log ϵ|)$. Apply the divergence theorem to the equation multiplied by ${\eta }_{ϵ}^{2}w$. The ordinary boundary terms vanish, and the Cauchy-Schwarz inequality givesEvery fixed compact subset eventually has ${\eta }_{ϵ}=1$, 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.