  In this paper, we present counterexamples showing that for any $p\in
(1,\infty)$, $p\neq 2$, there is a non-divergence form uniformly elliptic
operator with piecewise constant coefficients in $\mathbb{R}^2$ (constant on
each quadrant in $\mathbb{R}^2$) for which there is no $W^2_p$ estimate. The
corresponding examples in the divergence case are also discussed. One
implication of these examples is that the ranges of $p$ are sharp in the recent
results obtained in [4,5] for non-divergence type elliptic and parabolic
equations in a half space with the Dirichlet or Neumann boundary condition when
the coefficients do not have any regularity in a tangential direction.
