In this paper, we present counterexamples showing that for any $p\in(1,\infty)$, $p\neq 2$, there is a non-divergence form uniformly ellipticoperator with piecewise constant coefficients in $\mathbb{R}^2$ (constant oneach quadrant in $\mathbb{R}^2$) for which there is no $W^2_p$ estimate. Thecorresponding examples in the divergence case are also discussed. Oneimplication of these examples is that the ranges of $p$ are sharp in the recentresults obtained in [4,5] for non-divergence type elliptic and parabolicequations in a half space with the Dirichlet or Neumann boundary condition whenthe coefficients do not have any regularity in a tangential direction.
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机译：在本文中，我们提供了一些反例，表明对于任何$ p \ in（1，\ infty）$，$ p \ neq 2 $，在$ \ mathbb {R}中存在具有分段常数系数的非散度形式一致椭圆算子^ 2 $（$ \ mathbb {R} ^ 2 $中每个象限的常数），没有$ W ^ 2_p $的估算值。还讨论了分歧情况下的相应示例。这些例子的一个含义是，在系数[4,5]中，当系数不具有任何正则性时，在具有Dirichlet或Neumann边界条件的半空间中的非散度型椭圆和抛物方程的最新结果中，$ p $的范围非常大。沿切线方向。
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