= Weak formulation of the Robin Poisson problem
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{title2=$\int_\Omega\nabla u\cdot\nabla v+\int_{\partial\Omega}\beta uv=\int_\Omega fv$}
On a bounded <Lipschitz domain>, the <Poisson equation> $-\Delta u=f$ with <Robin boundary condition> $\partial_nu+\beta u=0$ has the displayed <weak formulation> in $H^1(\Omega)$ when $\beta$ is a bounded boundary coefficient. Boundary integrals use <Sobolev traces>. It follows from <Green's first identity>, requiring only first <weak derivatives> of $u$. If $\beta$ is uniformly positive, testing the difference of two solutions against itself proves uniqueness: its <gradient> and its boundary <Sobolev trace> vanish. The <Poincare inequality with a boundary trace> also makes this a <coercive bilinear form>.
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