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

We study partial Hölder regularity of the local minimizers u∈Wloc1,1(Ω;RN)$u\in W_{\mathrm{loc}}^{1,1}(\Omega;{\mathbb {R}^N})$ with N≥1$N\ge 1$ to the integral functional ∫ΩF(x,u,Du)dx$\int _\Omega F(x,u,Du)\,dx$ in a bounded domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$ for n≥2$n\ge 2$. Under the assumption of asymptotically convex to the borderline double‐phase functional ∫Ωb(x,u)|Du|p+a(x)|Du|plog(e+|Du|)dx,$$\begin{equation*} \hspace*{67pt}\int _\Omega b (x,u) {\left({|Du{|^p} + a(x)|Du{|^p}\log (e + |Du|)} \right)} \,dx, \end{equation*}$$where b(x,u)$b(x,u)$ satisfies VMO in x$x$ and is continuous in u$u$, respectively, and a(x)$a(x)$ 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 Ω0⊂Ω$ \Omega _0 \subset \Omega$ with Hn−p−εΩ∖Ω0=0$\mathcal {H}^{n-p-\varepsilon }\left(\Omega \backslash \Omega _0\right)=0$ for some small ε>0$ \varepsilon >0$, where Hs$\mathcal {H}^{s}$ denotes s$s$‐dimensional Hausdorff measure.