ABSTRACT

In this paper we obtain the interior Hölder regularity of the gradient for the elliptic $p\left(x\right)$-Laplacian equation with the logarithmic function $\begin{array}{rl}& \mathrm{d}\mathrm{i}\mathrm{v}\left({\left(A\mathrm{\nabla }u\cdot \mathrm{\nabla }u\right)}^{\frac{p\left(x\right)-2}{2}}\ln \left(e+{\left(A\mathrm{\nabla }u\cdot \mathrm{\nabla }u\right)}^{1/2}\right)A\mathrm{\nabla }u\right)\\ & =\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathbf{f}{|}^{p\left(x\right)-2}\ln \left(e+\left|\mathbf{f}\right|\right)\mathbf{f}\right)\end{array}$ under some proper assumptions on the Hölder continuous functions $p,\mathbf{f}$ and A. This extends previous results obtained by Giannetti and Passarelli di Napoli [Regularity results for a new class of functionals with non-standard growth conditions. J. Differ Equ. 2013;254(3):1280–1305], where A is the identity matrix and hence does not depend on x.

中文翻译：

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具有对数函数的椭圆 p(x)-拉普拉斯方程的 Hölder 估计

摘要

在本文中，我们获得了椭圆梯度的内部 Hölder 正则性$p\left(X\right)$- 具有对数函数的拉普拉斯方程$\begin{array}{rl}& \mathrm{d}\mathrm{一世}\mathrm{v}\left({\left(\mathrm{一个}\mathrm{\nabla }你\cdot \mathrm{\nabla }你\right)}^{\frac{p\left(X\right)-2}{2}}\ln \left(e+{\left(\mathrm{一个}\mathrm{\nabla }你\cdot \mathrm{\nabla }你\right)}^{1/2}\right)\mathrm{一个}\mathrm{\nabla }你\right)\\ & =\mathrm{d}\mathrm{一世}\mathrm{v}\left(|\mathbf{F}{|}^{p\left(X\right)-2}\ln \left(e+\left|\mathbf{F}\right|\right)\mathbf{F}\right)\end{array}$在对 Hölder 连续函数的一些适当假设下$p,\mathbf{F}$和A。这扩展了 Gianneti 和 Passarelli di Napoli 先前获得的结果[具有非标准生长条件的一类新泛函的规律性结果。J. 不同的 Equ。2013;254(3):1280–1305]，其中A是单位矩阵，因此不依赖于x。