Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth

Coscia, Alessandra; Mucci, Domenico

doi:10.1051/cocv:2002065

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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

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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 Zbl

DOI : 10.1051/cocv:2002065

Classification : 49J45, 49M20, 46E35
Keywords: integral representation, $\Gamma $-convergence, nonstandard growth conditions

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     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},
     mrnumber = {1925039},
     zbl = {1036.49022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2002065/}
}

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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

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