Partial regularity of minimizers of higher order integrals with (p, q)-growth

Abstract

We consider higher order functionals of the form

$F[u]=\int\limits_\Omega f(D^mu)\,{\rm d}x \qquad\text{for }u:\mathbb{R}^n\supset\Omega\to\mathbb{R}^N,$

where the integrand $f:{\textstyle \bigodot^m}(\R^{n},\R^{N})\to\mathbb{R}$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition.More precisely we assume that f fulfills the (p, q)-growth condition

\[\gamma|A|^p\le f(A)\le L(1+|A|^q)\qquad \mbox{for all }A \in {\textstyle \bigodot^m}(\R^{n},\R^{N}),\]

with γ, L > 0 and $1< p \le q<\min\big\{p+\frac1n,\frac{2n-1}{2n-2}p\big\}$. We study minimizers of thefunctional $F[\cdot]$ and prove a partial $C^{m,\alpha}_{\rm loc}$-regularity result.