Calculus of Variations and Geometric Measure Theory
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A. Davini

Integral representation of abstract functionals of autonomous type

created by davini on 07 Nov 2006
modified on 11 Jan 2009


Published Paper

Inserted: 7 nov 2006
Last Updated: 11 jan 2009

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Year: 2006


In this work we extend the results of M. Amar, G. Bellettini e S. Venturini (Proc. Roy. Soc. Edinburgh, 128A, pp. 193--217) to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties, and general integral growth conditions of superlinear type. We single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation, and sufficient as well to have an integral representation for its lower semicontinuous envelope. We also show that the integrand $F(x,q)$ satisfies some nice regularity properties in the $q$--variable, in particular a convexity--type property along lines. By adapting to the case at issue the reparametrization techniques introduced in a previous work, we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above, in particular it is closed by Gamma--convergence.

Keywords: relaxation, Gamma-convergence, Lagrangian minimizer


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