In this paper, we investigate the optimal $C^{1,\alpha }$ estimates for the elliptic $p\left(\cdot \right)$-Laplace equation: $div\left(a\left(x\right)\right|\nabla u{|}^{p\left(x\right)-2}\nabla u)=div\mathbf{h}\left(x\right)+f\left(x\right)\mathrm{in}\Omega$with $f\in L^{q\left(\cdot \right)}\left(\Omega \right)$ and $a,\mathbf{h}\in C^{\sigma }\left(\overline{\Omega }\right)$. Based on a certain geometric oscillation estimate, the scaling arguments and appropriate localization technique as well as the careful analysis on the variable exponents, we exhibit how the optimal Hölder exponent of $\nabla u$ is influenced by $p\left(\cdot \right)$, $q\left(\cdot \right)$ and $\sigma$. This work can be regarded as a natural follow up to the paper by Araújo and Zhang (in press).

中文翻译：

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在最佳状态 $C^{\mathrm{1个}，\alpha }$ 估计 $p（X）$-拉普拉斯类型方程

在本文中，我们研究了最优 $C^{\mathrm{1个}，\alpha }$ 椭圆的估计 $p（\cdot ）$-拉普拉斯方程： $div（\mathrm{一种}（X）|\nabla ü{|}^{p（X）-2}\nabla ü）=div\mathbf{H}（X）+F（X）\mathrm{在}\Omega$与 $F\in {\mathrm{大号}}^{q（\cdot ）}（\Omega ）$ 和 $\mathrm{一种}，\mathbf{H}\in C^{\sigma }（\overline{\Omega }）$。基于一定的几何振荡估计，缩放参数和适当的定位技术以及对可变指数的仔细分析，我们展示了最优的Hölder指数如何$\nabla ü$ 受...的影响 $p（\cdot ）$， $q（\cdot ）$ 和 $\sigma$。可以将这项工作视为对Araújo和Zhang（印刷中）论文的自然跟进。