= Power Caccioppoli inequality
{title2=$\int |Du|^2u^{\beta-2}\eta^2\leq\frac{C}{(\beta-1)^2}\int u^\beta|D\eta|^2$}
For a nonnegative bounded <weak subsolution> and $\beta>1$, test with $\eta^2(u+\varepsilon)^{\beta-1}$. Uniform ellipticity and <Cauchy-Schwarz inequality> give $\int |Du|^2u^{\beta-2}\eta^2\leq4\Lambda_*^2/[\lambda^2(\beta-1)^2]\int u^\beta|D\eta|^2$, with $\Lambda_*$ an <operator norm> bound for the coefficient <matrix>. The <Sobolev chain rule> and $\varepsilon\downarrow0$ justify fractional powers and zero values. An entrywise bound $\Lambda$ yields $\Lambda_*\leq n\Lambda$.
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