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

In this paper, we present counterexamples showing that for any $pin (1,infty)$, $p eq 2$, 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.

Boundary value problems for parabolic operators in a time-varying domain

We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.