Regularity for multi-phase variational problems

We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under sharp assumptions relating the couples $(p,q)$ and $(p,s)$ to the H\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$, respectively.