Partial regularity of minimizers of higher order integrals with $\left(p,q\right)$-growth
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 472-492.

We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}u:{ℝ}^{n}\supset \Omega \to {ℝ}^{N},$ where the integrand $f:{⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right)\to ℝ$ , m 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition

 ${\gamma |A|}^{p}\le f\left(A\right)\le {L\left(1+|A|}^{q}\right)\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}A\in {⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right),$
with γ, L > 0 and $1. We study minimizers of the functional $F\left[·\right]$ and prove a partial ${C}_{\mathrm{loc}}^{m,\alpha }$-regularity result.

DOI: 10.1051/cocv/2010016
Classification: 49N60,  49N99,  49J45
Keywords: higher order functionals, non-standard growth, regularity theory
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Schemm, Sabine. Partial regularity of minimizers of higher order integrals with $(p, q)$-growth. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 472-492. doi : 10.1051/cocv/2010016. http://www.numdam.org/articles/10.1051/cocv/2010016/

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