= No uniformly positive coefficient in a zero-boundary Sobolev space
On a nonempty bounded domain where the <Poincare inequality> holds, no $a\in H_0^1(\Omega)$ can satisfy $a\geq a_->0$ almost everywhere. For $g(s)=\min(\max(s,0),a_-)$, <Lipschitz truncation preserves zero-boundary Sobolev spaces>, so $g(a)\in H_0^1(\Omega)$. But $g(a)$ would be the nonzero constant $a_-$, whose zero gradient contradicts the <Poincare inequality>. In a uniformly elliptic <divergence-form elliptic operator>, the homogeneous boundary condition belongs to the unknown and test functions, not to a coefficient bounded below by a positive constant.
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