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

TL;DR

This paper constructs counterexamples demonstrating that for certain elliptic operators with piecewise constant coefficients, $W^2_p$ estimates fail for all $p$ except 2, highlighting limitations of regularity results in such settings.

Contribution

It provides the first explicit counterexamples showing the failure of $W^2_p$ estimates for elliptic equations with piecewise constant coefficients in $\,\mathbb{R}^2$ for all $p \neq 2$, clarifying the sharpness of recent regularity results.

Findings

Counterexamples for $p \neq 2$ show no $W^2_p$ estimates exist.

Examples in both non-divergence and divergence forms are constructed.

Results confirm the sharpness of previous regularity theorems.

Abstract

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