We prove a $C^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left(D^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand f satisfies (p,q) growth conditions.


We consider higher order functionals of the form$F\left[u\right]=\underset{\Omega }{\int }f\left(D^{m}u\right)\mathrm{d}x\text{for}u:{ℝ}^n\supset \Omega \to {ℝ}^N,$ where the integrand $f:{⨀}^m({ℝ}^n,{ℝ}^N)\to ℝ$, m≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^p\le f\left(A\right)\le {L(1+|A|}^q)\text{for}\text{all}A\in {⨀}^m({ℝ}^n,{ℝ}^N),$$with γ, L > 0 and $1<p\le q<min\{p+\frac{1}{n},\frac{2n-1}{2n-2}p\}$. We study minimizers of the functional $F[·]$ and prove a partial $C_{\mathrm{loc}}^{m,\alpha }$-regularity result.

We consider higher order functionals of the form$F\left[u\right]=\underset{\Omega }{\int }f\left(D^{m}u\right)\mathrm{d}x\text{for}u:{ℝ}^n\supset \Omega \to {ℝ}^N,$ where the integrand $f:{⨀}^m({ℝ}^n,{ℝ}^N)\to ℝ$, m≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^p\le f\left(A\right)\le {L(1+|A|}^q)\text{for}\text{all}A\in {⨀}^m({ℝ}^n,{ℝ}^N),$$with γ, L > 0 and $1<p\le q<min\{p+\frac{1}{n},\frac{2n-1}{2n-2}p\}$. We study minimizers of the functional $F[·]$ and prove a partial $C_{\mathrm{loc}}^{m,\alpha }$-regularity result.