Regularity for multi-phase problems at nearly linear growth

Abstract

Minima of the log-multiphase variational integral $$w\mapsto {\int }_{\Omega }\left[|Dw|\log (1+|Dw|)+a(x)|Dw{|}^q+b(x)|Dw{|}^s\right]\mathrm{d}x,$$ have locally Hölder continuous gradient under sharp quantitative bounds linking the growth powers $(q,s)$ to the Hölder exponents of the modulating coefficients $a(\cdot )$ and $b(\cdot )$ respectively.