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Minima of the log-multiphase variational integral $$  w \mapsto \int_{\Omega} \left[|Dw|\log(1+|Dw|) + a(x)|Dw|^q +b(x)|Dw|^s\right] \, {\rm d}x\,, $$ have locally H\"older continuous gradientunder sharp quantitative bounds linking the growth powers $(q,s)$ to theH\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$respectively.

Regularity for multi-phase problems at nearly linear growth