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

where

\Omega\subseteq\mathbb{R}^n Ω ⊆ 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} f  ⁣ : Ω × R N × R N × n → R

is not necessarily continuous in any of its three arguments. In particular, due to the use of a suitable asymptotic relatedness condition,

f f

possesses continuity and convexity only as the norm of its third argument tends to infinity. Since, in particular,

v 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.