Double phase quasiconvex functionals and their partial regularity theory

We consider degenerate nonautonomous energies $${\int }_{\mathrm{\Omega }}f(x,Dv)dx,$$ for vector-valued functions $v\in W^{1,1}(\mathrm{\Omega },{\mathbb{R}}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function $H(x,t)=t^p+a(x)t^q$. We establish partial Hölder regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on $H$ and $f$. Our approach relies on two approximation results: $\mathcal{A}$-harmonic approximation and a variational version of the $\phi$-harmonic approximation.