On the impossibility of W-p(2) estimates for elliptic equations with piecewise constant coefficients

摘要

In this paper, we present counterexamples showing that for any p is an element of (1, infinity), p not equal 2, there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in R-2 (constant on each quadrant in R-2) for which there is no W-p(2) 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. (C) 2014 Elsevier Inc. All rights reserved.