Integral representation and <span class="mathjax-formula">$\sf \Gamma $</span>-convergence of variational integrals with <span class="mathjax-formula">${p(x)}$</span>-growth

Coscia, Alessandra; Mucci, Domenico

doi:10.1051/cocv:2002065

Integral representation and

Γ

-convergence of variational integrals with

p (x)

-growth

Coscia, Alessandra ; Mucci, Domenico

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 495-519.

Résumé

We study the integral representation properties of limits of sequences of integral functionals like $\int f (x, D u) d x$ under nonstandard growth conditions of $(p, q)$ -type: namely, we assume that

{| z |}^{p (x)} \leq f (x, z) \leq {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, D u)

depending explicitly on

u

; finally we weaken the assumption allowing

p (x)

to be discontinuous on nice sets.

MR 1925039 | Zbl 1036.49022

DOI : https://doi.org/10.1051/cocv:2002065
Classification : 49J45, 49M20, 46E35
Mots clés : integral representation,

Γ

-convergence, nonstandard growth conditions

BibTeX
RIS

@article{COCV_2002__7__495_0,
     author = {Coscia, Alessandra and Mucci, Domenico},
     title = {Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {495--519},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002065},
     zbl = {1036.49022},
     mrnumber = {1925039},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002065/}
}

TY  - JOUR
AU  - Coscia, Alessandra
AU  - Mucci, Domenico
TI  - Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
DA  - 2002///
SP  - 495
EP  - 519
VL  - 7
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002065/
UR  - https://zbmath.org/?q=an%3A1036.49022
UR  - https://www.ams.org/mathscinet-getitem?mr=1925039
UR  - https://doi.org/10.1051/cocv:2002065
DO  - 10.1051/cocv:2002065
LA  - en
ID  - COCV_2002__7__495_0
ER  -

Coscia, Alessandra; Mucci, Domenico. Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 495-519. doi : 10.1051/cocv:2002065. http://www.numdam.org/articles/10.1051/cocv:2002065/

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