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

Abstract

We study vector-valued local minimizers of the borderline double phase functional whose integrand is $|D\mathbf{w}|^p+a(x)|D\mathbf{w}|^p\log^γ(e+|D\mathbf{w}|)$, where $p>1$, $γ>0$ and the coefficient $a(\cdot)$ is nonnegative and bounded. If the modulus of continuity $ω(\cdot)$ of $a(\cdot)$ satisfies $ω(r)\log^γ(1/r)\to0$ as $r\to0$, 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 $γ=1$. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.