Homogeneous divergence-form elliptic equation ($\mathrm{div}(A\nabla u)=0$)

For a bounded measurable coefficient matrix $A$, an $H^1$ weak solution of $\mathrm{div}(A\nabla u)=0$ satisfies $\int A\nabla u\cdot \nabla \phi =0$ 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 $A$.