Gradient regularity for non-autonomous functionals with Dini or non-Dini continuous coefficients

We prove $C^1$ regularity for local vectorial minimizers of the non-autonomous functional \[ w\in W^{1,1}_{\rm loc}(\Omega;\R^N)\longmapsto \int_{\Omega}b(x)\big[|Dw|^p+a(x)|Dw|^p\log(e+|Dw|)\big] \,dx\,, \] with $\Omega$ open subset of $\R^n$, $n\geq2$ , $p>1$, $0\leq a(\cdot)\leq \|a\|_{L^{\infty}(\Omega)}<\infty$ and $0<\nu\leq b(\cdot)\leq L$. The result is obtained provided that the function $a(\cdot)$ is $\log$-Dini continuous and that the coefficient $b(\cdot)$ is Dini continuous or it is weakly differentiable and its gradient locally belongs to the Lorentz space $L^{n,1}(\Omega;\R^n)$.