 Failure of L2-to-Linfinity Poisson gradient bounds in three dimensions

Inside any three-dimensional domain containing a ball, let $u_{\epsilon }(x)={\epsilon }^{1/2}\varphi ((x-x_0)/\epsilon )$ for a fixed smooth compactly supported nonconstant function. The right-hand side $f_{\epsilon }=\Delta u_{\epsilon }$ has constant $L^2$ norm, whereas $\Vert \nabla u_{\epsilon }{\Vert }_{\infty }={\epsilon }^{-1/2}\Vert \nabla \varphi {\Vert }_{\infty }$ diverges. The functions have zero boundary trace, giving a scaling counterexample to a uniform $L^2$-forcing gradient estimate.