= Poisson uniqueness with a nonnegative Robin normal coefficient
{c}
On a bounded region, two sufficiently regular solutions of the <Poisson equation> with the same forcing and boundary condition $\alpha\partial_n\phi+\phi=\gamma$, where $\alpha\ge0$, have harmonic difference $w$. <Green's first identity> gives $\int|\nabla w|^2=\int_{\partial V}w\partial_nw=-\int_{\partial V}\alpha(\partial_nw)^2\le0$. Hence $w$ is constant on each component, and the boundary condition makes that constant zero. The proof never divides by $\alpha$, so it includes variable coefficients and Dirichlet portions where $\alpha=0$.
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