Calculus of Variations and Geometric Measure Theory

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