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

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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 W²ₚ 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.