Hölder estimates for the elliptic p(x)-Laplacian equation with the logarithmic function

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.

Keywords:

• $C^{1\alpha }$
• Hölder
• gradient
• regularity
• divergence
• elliptic
• $p\left(x\right)$-Laplacian
• logarithmic

2010 Mathematics Subject Classifications:

• 35J60
• 35J70

Disclosure statement

No potential conflict of interest was reported by the author(s).