Regularity for vectorial double phase functionals with two modulating coefficients and logarithmic-type growth

Abstract

We study vector-valued local minimizers of double phase functionals with two modulating coefficients and logarithmic-type growth. The energy density is $a(x)|Du{|}^p+b(x)|Du{|}^p\log (e+|Du|)$, where the nonnegative coefficients $a(\cdot )$ and $b(\cdot )$ may vanish but their sum is bounded away from zero. Assuming that $a(\cdot )$ is uniformly continuous and that $b(\cdot )$ satisfies a vanishing log-Hölder continuity condition, we prove that minimizers are locally Hölder continuous for every exponent in $(0,1)$. If both coefficients are Hölder continuous, then the gradient of a minimizer is locally Hölder continuous. The proof combines Gehring-type higher integrability with a comparison argument.