Partial Hölder regularity for asymptotically convex functionals with borderline double-phase growth

We study partial Hölder regularity of the local minimizers $$u\in W_{\mathrm{loc}}^{1,1}(\Omega;{\mathbb {R}^N})$$ with $$N\ge 1$$ to the integral functional $$\int _\Omega F(x,u,Du)\,dx$$ in a bounded domain $$\Omega \subset \mathbb {R}^n$$ for $$n\ge 2$$. Under the assumption of asymptotically convex to the borderline double-phase functional

where $$b(x,u)$$ satisfies VMO in $$x$$ and is continuous in $$u$$, respectively, and $$a(x)$$ is a strongly log-Hölder continuous function, we prove that the local minimizer of such a functional is locally Hölder continuous with an explicit Hölder exponent in an open set $$ \Omega _0 \subset \Omega$$ with $$\mathcal {H}^{n-p-\varepsilon }\left(\Omega \backslash \Omega _0\right)=0$$ for some small $$ \varepsilon >0$$, where $$\mathcal {H}^{s}$$ denotes $$s$$-dimensional Hausdorff measure.