= Coercive Neumann energy with a positive reaction term
{title2=$B[u,u]\ge\min(\nu,\mu)\|u\|_{H^1}^2$}
A symmetric uniformly elliptic divergence-form energy with zeroth-order coefficient at least $\mu>0$ controls the full $H^1$ <norm>, including constants. Its quadratic energy minus a <bounded> load has a unique <minimizer> by the direct method, and vanishing <first variation> gives the natural <Neumann boundary condition>[Neumann] weak equation. Unlike a pure <Neumann boundary condition>[Neumann] <Laplacian>, this problem needs no zero-mean compatibility condition. The exact energy difference from the <weak solution> is half the bilinear energy of the difference.
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