Integral representation and Γ-convergence of variational integrals with p(x)-growth
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 495-519

We study the integral representation properties of limits of sequences of integral functionals like ∫f(x,Du)dx 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.

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},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {7},
     doi = {10.1051/cocv:2002065},
     mrnumber = {1925039},
     zbl = {1036.49022},
     language = {en},
     url = {https://www.numdam.org/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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