= Homogeneous divergence-form elliptic equation
{title2=$\operatorname{div}(A\nabla u)=0$}
For a bounded measurable coefficient <matrix> $A$, an $H^1$ <weak solution> of $\operatorname{div}(A\nabla u)=0$ satisfies $\int A\nabla u\cdot\nabla\varphi=0$ for every compactly supported smooth <test function>. By density this identity extends to compactly supported $H^1$ tests. If $A$ is symmetric and $\lambda I\le A\le\Lambda I$, <Caccioppoli inequalities> follow without differentiating $A$.
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