In this note we are concerned with interior regularity properties of the p-Poisson problem ${\mathrm{\Delta }}_p(u)=f$ with $p>2$. For all $0<\lambda \le 1$ we constuct right-hand sides f of differentiability $-1+\lambda$ such that the (Besov-) smoothness of corresponding solutions u is essentially limited to $1+\lambda /(p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152 (1998), 176–201], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130 (2016), 298–329].

中文翻译：

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关于 p>2 的 p-Poisson 问题缺乏内部规律性

在本笔记中，我们关注p- Poisson 问题的内部规律性${\mathrm{\Delta }}_{磷}(你)=F$ 和 $磷>2$. 对所有人$0<\lambda \le 1$我们constuct右手边˚F微性$-1+\lambda$使得对应解u的 (Besov-) 平滑度基本上限于$1+\lambda /(磷-1)$. 这些陈述是局部性质的，涵盖了所有可积性参数。它们特别暗示了由于 Savaré [J. 功能。肛门。152 (1998), 176-201]，以及 Dahlke 等人最近的一些 Besov 正则性结果。[非线性肛门。130 (2016), 298–329]。