Borderline double phase functionals with powers of the logarithm: regularity in the vectorial case

We study vector-valued local minimizers of the borderline double phase functional whose integrand is $\mathrm{\mid }D\mathbf{w}{\mathrm{\mid }}^p+a(x)\mathrm{\mid }D\mathbf{w}{\mathrm{\mid }}^p{\log }^{\gamma }(e+\mathrm{\mid }D\mathbf{w}\mathrm{\mid })$, where $p>1$, $\gamma >0$ and the coefficient $a(\cdot )$ is nonnegative and bounded. If the modulus of continuity $\omega (\cdot )$ of $a(\cdot )$ satisfies $\omega (r){\log }^{\gamma }(1\mathrm{/}r)\to 0$ as $r\to 0$, then minimizers are locally Hölder continuous for every exponent less than one. If $a(\cdot )$ is Hölder continuous, then the gradients of minimizers are locally Hölder continuous. In the vector-valued setting these results were previously available only for $\gamma =1$. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.