Everywhere regularity for local minimizers of asymptotically convex non-autonomous functionals

We consider everywhere regularity of local vectorial minimizers $u:\Omega \to R^N(N\ge 2)$ to a class of non-autonomous functionals $J(u,\Omega )={\int }_{\Omega }\Phi \left(a\left(x\right)\left|Du\right|\right)dx,$where $\Omega$ is a bounded open subset of $R^n(n\ge 2)$. Under assumptions that $\Phi$ is an Orlicz function and the coefficient function $a:\Omega \to R$ belongs to $VMO\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$, we prove that every local minimizer of such functional is an everywhere Hölder continuity with any Hölder exponent $\alpha \in (0,1)$ by employing a perturbation method, hole filling technique and the iteration lemma. Furthermore, if the coefficient $a\in C_{loc}^{0,\kappa }\left(\Omega \right)$ with $\kappa \in (0,1)$ is satisfied, then a local everywhere Hölder continuity of the gradient is derived for such local minimizer. As a generalization of our main theorem, the same regularity conclusion holds for an asymptotically convex functional as follows: $F(u,\Omega )={\int }_{\Omega }f(x,Du)dx,$where the integrand $f(x,Du)$ is only asymptotically regular with respect to the integrand $\Phi \left(a\left(x\right)\left|Du\right|\right)$.