= Failure of L2-to-Linfinity Poisson gradient bounds in three dimensions
Inside any three-dimensional domain containing a ball, let $u_\varepsilon(x)=\varepsilon^{1/2}\phi((x-x_0)/\varepsilon)$ for a fixed smooth compactly supported nonconstant function. The right-hand side $f_\varepsilon=\Delta u_\varepsilon$ has constant $L^2$ <norm>, whereas $\|\nabla u_\varepsilon\|_\infty=\varepsilon^{-1/2}\|\nabla\phi\|_\infty$ diverges. The functions have zero boundary trace, giving a scaling counterexample to a uniform $L^2$-forcing <gradient> estimate.
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