On the impossibility of $W_p^2$ estimates for elliptic equations with piecewise constant coefficients

Abstract

In this paper, we present counterexamples showing that for any $p\in (1,\infty )$, $p\ne 2$, 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 $W_p^2$ 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.

arXiv comment10 pages