Superlevel Caccioppoli inequality ($\int {\phi }^2|\nabla (u-k{)}^{+}{|}^2\le C\int ((u-k{)}^{+}{)}^2|\nabla \phi {|}^2$)

For a weak solution of $-\mathrm{div}(A\nabla u)=0$ with real symmetric coefficients satisfying the two-sided uniform ellipticity bounds $\lambda I\le A\le \Lambda I$, the displayed Caccioppoli inequality holds with $C=4{\Lambda }^2/{\lambda }^2$ for a smooth cutoff function $\phi$ of compact support. Put $w=(u-k{)}^{+}$ and test the weak formulation with ${\phi }^{2}w$. The Sobolev chain rule gives $\nabla w=1_{\{u>k\}}\nabla u$, and Cauchy-Schwarz inequality yields $\lambda \int {\phi }^2|\nabla w{|}^2\le 2\Lambda (\int {\phi }^2|\nabla w{|}^2{)}^{1/2}(\int w^2|\nabla \phi {|}^2{)}^{1/2}$. Divide and square. The constant is independent of the level $k$.