Gradient Regularity for the Solutions to p(<middle dot>)-Laplacian Equations

In the present paper, we propose the investigation of variable exponent p-Laplace equations with logarithmic growth in divergence form. Under weak regularity assumptions on the boundary of domains and the exponent p(<middle dot>), the global gradient bounds in norms for solutions are well-established in a class of generalized function spaces via the presence of fractional maximal operators M-alpha. This effort can be developed in the theory of energy functionals satisfying certain nonstandard growth conditions, including problems governed by the p(<middle dot>)-Laplacian operator.