In this paper we produce new, optimal, regularity results for the solutions to $p$-Poisson equations. We argue through a delicate approximation method, under a smallness regime for the exponent $p$, that imports information from a limiting profile driven by the Laplace operator. Our arguments contain a novelty of technical interest, namely a sequential stability result; it connects the solutions to $p$-Poisson equations with harmonic functions, yielding improved regularity for the former. Our findings relate a smallness regime with improved $\mathcal{C}^{1,1-}$-estimates in the presence of $L^\infty$-source terms.

中文翻译：

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改进了 p-Poisson 方程的正则性

在本文中，我们为 $p$-Poisson 方程的解产生了新的、最优的、正则性的结果。我们通过一种微妙的近似方法进行论证，在指数 $p$ 的小范围内，从由拉普拉斯算子驱动的限制分布中导入信息。我们的论点包含技术兴趣的新颖性，即顺序稳定性结果；它将解与具有调和函数的 $p$-Poisson 方程联系起来，为前者产生改进的规律性。我们的研究结果与在 $L^\infty$ 源项存在下改进的 $\mathcal{C}^{1,1-}$-estimates 相关的小型机制。