= Superlevel Caccioppoli inequality
{title2=$\int\varphi^2|\nabla(u-k)^+|^2\leq C\int((u-k)^+)^2|\nabla\varphi|^2$}
For a <weak solution> of $-\operatorname{div}(A\nabla u)=0$ with real symmetric coefficients satisfying the two-sided <uniform ellipticity> bounds $\lambda I\leq A\leq\Lambda I$, the displayed <Caccioppoli inequality> holds with $C=4\Lambda^2/\lambda^2$ for a smooth <cutoff function> $\varphi$ of <compact support>. Put $w=(u-k)^+$ and test the <weak formulation> with $\varphi^2w$. The <Sobolev chain rule> gives $\nabla w=\mathbf1_{\{u>k\}}\nabla u$, and <Cauchy-Schwarz inequality> yields $\lambda\int\varphi^2|\nabla w|^2\leq2\Lambda(\int\varphi^2|\nabla w|^2)^{1/2}(\int w^2|\nabla\varphi|^2)^{1/2}$. Divide and square. The constant is independent of the level $k$.
Back to article page
