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

In this paper we obtain the interior Hölder regularity of the gradient for the elliptic p (x) -Laplacian equation with the logarithmic function d i v A ∇ u ⋅ ∇ u p (x) − 2 2 ln e + A ∇ u ⋅ ∇ u 1 / 2 A ∇ u = d i v | f | p (x) − 2 ln e + f f under some proper assumptions on the Hölder continuous functions p , 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. [ABSTRACT FROM AUTHOR]