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

Alessandra CosciaDomenico Mucci — 2002

ESAIM: Control, Optimisation and Calculus of Variations

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 , D u ) depending explicitly on u ; finally we weaken the assumption allowing p ( x ) to be discontinuous on nice sets.

Integral representation and Γ-convergence of variational integrals with -growth

Alessandra CosciaDomenico Mucci — 2010

ESAIM: Control, Optimisation and Calculus of Variations

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 -type: namely, we assume that | z | p ( x ) f ( x , z ) L ( 1 + | z | p ( x ) ) . Under weak assumptions on the continuous function , we prove -convergence to integral functionals of the same type. We also analyse the case of integrands depending explicitly on ; finally we weaken the assumption allowing to be discontinuous on nice sets.

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