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

Sabine Schemm

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 472-492
  • ISSN: 1292-8119

Abstract

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We consider higher order functionals of the form F [ u ] = ∫ Ω f ( D m u ) d x for u : ℝ n ⊃ Ω → ℝ N , where the integrand f : ⨀ m ( ℝ n , ℝ N ) → 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 γ | A | p ≤ f ( A ) ≤ L ( 1 + | A | q ) for all A ∈ ⨀ m ( ℝ n , ℝ N ) , with γ, L > 0 and 1 < p ≤ q < min { p + 1 n , 2 n - 1 2 n - 2 p } . We study minimizers of the functional F [ · ] and prove a partial C loc m , α -regularity result.

How to cite

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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 17.2 (2011): 472-492. <http://eudml.org/doc/276327>.

@article{Schemm2011,
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 the functional $F[\cdot]$ and prove a partial $C^\{m,\alpha\}_\{\rm loc\}$-regularity result. },
author = {Schemm, Sabine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Higher order functionals; non-standard growth; regularity theory; higher order functionals},
language = {eng},
month = {5},
number = {2},
pages = {472-492},
publisher = {EDP Sciences},
title = {Partial regularity of minimizers of higher order integrals with (p, q)-growth},
url = {http://eudml.org/doc/276327},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Schemm, Sabine
TI - Partial regularity of minimizers of higher order integrals with (p, q)-growth
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 472
EP - 492
AB - 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 the functional $F[\cdot]$ and prove a partial $C^{m,\alpha}_{\rm loc}$-regularity result.
LA - eng
KW - Higher order functionals; non-standard growth; regularity theory; higher order functionals
UR - http://eudml.org/doc/276327
ER -

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