Partial Hölder Continuity of Minimizers of Functionals Satisfying a General Asymptotic Relatedness Condition

We consider the partial Hölder continuity of minimizers of functionals of the form $$v\mapsto\int_{\Omega}f(x,v,Dv)\ dx,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. In our setting the integrand $f\colon \Omega\times\mathbb{R}^N\times \mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is not necessarily continuous in any of its three arguments. In particular, due to the use of a suitable asymptotic relatedness condition, $f$ possesses continuity and convexity only as the norm of its third argument tends to infinity. Since, in particular, $v$ is possibly vector-valued, this provides a generalization of certain existing regularity results in the literature and helps to further build a low-order regularity theory.