Regularity results for non-autonomous functionals with $$varvec{{L}log {L}}$$-growth and Orlicz Sobolev coefficients

We study the regularity properties of local minimizers of non-autonomous convex integral functionals of the type

$$begin{aligned} mathcal {F}( u, Omega )= int _{Omega } ! f(x,Du) , ,dx, end{aligned}$$

when the integrand f has almost linear growth with respect to the gradient variable and the dependence on the x-variable is controlled by a function which belongs to a suitable Orlicz Sobolev space.