New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński; Agnieszka Kałamajska

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 12, Issue: 1, page 64-92
  • ISSN: 1292-8119

Abstract

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We consider the lower semicontinuous functional of the form I f ( u ) = ∫ Ω f ( u ) d x where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.

How to cite

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Chełmiński, Krzysztof, and Kałamajska, Agnieszka. "New convexity conditions in the calculus of variations and compensated compactness theory." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2005): 64-92. <http://eudml.org/doc/90791>.

@article{Chełmiński2005,
abstract = { We consider the lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u)\{\rm d\}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions. },
author = {Chełmiński, Krzysztof, Kałamajska, Agnieszka},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Quasiconvexity; rank-one convexity; semicontinuity.; semicontinuity},
language = {eng},
month = {12},
number = {1},
pages = {64-92},
publisher = {EDP Sciences},
title = {New convexity conditions in the calculus of variations and compensated compactness theory},
url = {http://eudml.org/doc/90791},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Chełmiński, Krzysztof
AU - Kałamajska, Agnieszka
TI - New convexity conditions in the calculus of variations and compensated compactness theory
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2005/12//
PB - EDP Sciences
VL - 12
IS - 1
SP - 64
EP - 92
AB - We consider the lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u){\rm d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.
LA - eng
KW - Quasiconvexity; rank-one convexity; semicontinuity.; semicontinuity
UR - http://eudml.org/doc/90791
ER -

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