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.
|Number of pages||12|
|Journal||Journal of Functional Analysis|
|Publication status||Published - 2014 Nov 15|
- Elliptic equations with piecewise constant coefficients
- Wp2 estimates
ASJC Scopus subject areas