Lower semicontinuity of multiple μ -quasiconvex integrals

Ilaria Fragalà

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 105-124
  • ISSN: 1292-8119

Abstract

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Lower semicontinuity results are obtained for multiple integrals of the kind ∫ ℝ n f ( x , ∇ μ u ) d μ , where μ is a given positive measure on ℝ n , and the vector-valued function u belongs to the Sobolev space H μ 1 , p ( ℝ n , ℝ m ) associated with μ . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ . More precisely, for fully general μ , a notion of quasiconvexity for f along the tangent bundle to μ , turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.

How to cite

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Fragalà, Ilaria. "Lower semicontinuity of multiple $\sf \mu $-quasiconvex integrals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 105-124. <http://eudml.org/doc/245959>.

@article{Fragalà2003,
abstract = {Lower semicontinuity results are obtained for multiple integrals of the kind $\int _\{\mathbb \{R\}^n\}\!f(x, \!\nabla _\mu u\!)\{\rm d\} \mu $, where $\mu $ is a given positive measure on $\mathbb \{R\}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H ^\{1,p\}_\mu (\mathbb \{R\}^n, \mathbb \{R\}^m)$ associated with $\mu $. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu $. More precisely, for fully general $\mu $, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu $, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when $\mu $ belongs to a suitable class of rectifiable measures.},
author = {Fragalà, Ilaria},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Borel measures; tangent properties; lower semicontinuity},
language = {eng},
pages = {105-124},
publisher = {EDP-Sciences},
title = {Lower semicontinuity of multiple $\sf \mu $-quasiconvex integrals},
url = {http://eudml.org/doc/245959},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Fragalà, Ilaria
TI - Lower semicontinuity of multiple $\sf \mu $-quasiconvex integrals
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 105
EP - 124
AB - Lower semicontinuity results are obtained for multiple integrals of the kind $\int _{\mathbb {R}^n}\!f(x, \!\nabla _\mu u\!){\rm d} \mu $, where $\mu $ is a given positive measure on $\mathbb {R}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H ^{1,p}_\mu (\mathbb {R}^n, \mathbb {R}^m)$ associated with $\mu $. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu $. More precisely, for fully general $\mu $, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu $, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when $\mu $ belongs to a suitable class of rectifiable measures.
LA - eng
KW - Borel measures; tangent properties; lower semicontinuity
UR - http://eudml.org/doc/245959
ER -

References

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