= Semilinear Poisson contraction threshold
{title2=$\epsilon C_P^2<1$}
For a globally <Lipschitz continuity>[Lipschitz] nonlinearity of constant $\epsilon$, the map taking $v$ to the <Dirichlet boundary condition>[zero-Dirichlet] Poisson solution with load $f(v)$ is a contraction in the <Sobolev norm>[gradient norm] if the displayed inequality holds. <Poincaré inequality>[Poincaré] gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear <Poisson equation> gives another factor $C_P$ back to the <Sobolev norm>[gradient norm]. The condition yields a unique <weak solution> without requiring $f(0)=0$. With the optimal <Poincaré inequality>[Poincaré] constant it reads $\epsilon<\lambda_1$.
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