Lower semicontinuity of multiple µ-quasiconvex integrals

Ilaria Fragalà

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 µ-quasiconvex integrals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 105-124. <http://eudml.org/doc/90683>.

@article{Fragalà2010,
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 μ 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 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. },
author = {Fragalà, Ilaria},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Borel measures; tangent properties; lower semicontinuity.; lower semicontinuity},
language = {eng},
month = {3},
pages = {105-124},
publisher = {EDP Sciences},
title = {Lower semicontinuity of multiple µ-quasiconvex integrals},
url = {http://eudml.org/doc/90683},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Fragalà, Ilaria
TI - Lower semicontinuity of multiple µ-quasiconvex integrals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 μ 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 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.
LA - eng
KW - Borel measures; tangent properties; lower semicontinuity.; lower semicontinuity
UR - http://eudml.org/doc/90683
ER -

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