Integral representation and Γ-convergence of variational integrals with p(x)-growth

Alessandra Coscia; Domenico Mucci

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

  • Volume: 7, page 495-519
  • ISSN: 1292-8119

Abstract

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We study the integral representation properties of limits of sequences of integral functionals like   ∫ f ( x , D u ) d x   under nonstandard growth conditions of (p,q)-type: namely, we assume that | z | p ( x ) ≤ f ( x , z ) ≤ L ( 1 + | z | p ( x ) ) . Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.

How to cite

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Coscia, Alessandra, and Mucci, Domenico. "Integral representation and Γ-convergence of variational integrals with p(x)-growth." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 495-519. <http://eudml.org/doc/90633>.

@article{Coscia2010,
abstract = { We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\,\{\rm d\}x$  under nonstandard growth conditions of (p,q)-type: namely, we assume that $$ \vert z\vert^\{p(x)\}\leq f(x,z)\leq L(1+\vert z\vert^\{p(x)\})\,. $$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets. },
author = {Coscia, Alessandra, Mucci, Domenico},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Integral representation; Γ-convergence; nonstandard growth conditions.; integral representation; Gamma-convergence; nonstandard growth conditions; integral functionals},
language = {eng},
month = {3},
pages = {495-519},
publisher = {EDP Sciences},
title = {Integral representation and Γ-convergence of variational integrals with p(x)-growth},
url = {http://eudml.org/doc/90633},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Coscia, Alessandra
AU - Mucci, Domenico
TI - Integral representation and Γ-convergence of variational integrals with p(x)-growth
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 495
EP - 519
AB - We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\,{\rm d}x$  under nonstandard growth conditions of (p,q)-type: namely, we assume that $$ \vert z\vert^{p(x)}\leq f(x,z)\leq L(1+\vert z\vert^{p(x)})\,. $$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.
LA - eng
KW - Integral representation; Γ-convergence; nonstandard growth conditions.; integral representation; Gamma-convergence; nonstandard growth conditions; integral functionals
UR - http://eudml.org/doc/90633
ER -

References

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