We show that local minimizers of the non-autonomous functional \(\begin{aligned} {\mathcal {P}}_{\log }(u,\Omega )= \int _\Omega |Du|^p\big (1+a(x)\log \left( e+|Du|\right) \big )\,dx,\qquad p>1, \end{aligned}\) have continuous gradient provided that the function \(a(\cdot )\) is (almost everywhere) non-negative and weakly differentiable, and moreover its gradient locally belongs to the Lorentz-Zygmund space \(L^{n,1}\log L\) . This gives a precise insight of the fact that for this type of two-phase functionals the lack of uniform ellipticity can be overcome by additional regularity of the switching coefficient \(a(\cdot )\) ; the novelty is that the condition is not pointwise, but has integral character, and actually improves the known results ensuring regularity for minimizers of such functionals.
