= Radial Poisson gradient estimate
{title2=$\|\nabla u\|_\infty\le C\|f\|_2$}
For a radial Dirichlet solution on a fixed three-dimensional shell $a<r<b$, $(r^2U')'=r^2F$. Thus $r^2U'=C_0+\int_a^rs^2F(s)\,ds$, and $U(a)=U(b)=0$ bounds $|C_0|$ by the supremum of that integral. <Cauchy-Schwarz inequality> in the radial volume measure bounds the integral by $C\|f\|_2$. Since $r\ge a>0$, the displayed <gradient> bound follows. The estimate exploits radiality and does not extend to arbitrary forcing in three dimensions.
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