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

Schemm, Sabine

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 472-492.

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 $1<p\le q<min\left\{p+\frac{1}{n},\frac{2n-1}{2n-2}p\right\}$. We study minimizers of the functional $F[·]$ and prove a partial $C_{\mathrm{loc}}^{m,\alpha }$-regularity result.

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