On the impossibility of Wp2 estimates for elliptic equations with piecewise constant coefficients

In this paper, we present counterexamples showing that for any p∈(1,∞), p≠2, there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in R2 (constant on each quadrant in R2) for which there is no Wp2 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.