The relaxation of some classes of variational integrals with pointwise continuous-type gradient constraints

摘要

Relaxation problems for a functional of the type G(u) = ∫&UOmega; g(x, &DEL; u)dx are analyzed, where &UOmega; is a bounded smooth subset of R-N and g is a Caratheodory function, when the admissible u are forced to satisfy a pointwise gradient constraint of the type &DEL; u(x) ∈ C(x) for a.e. x ∈ &UOmega;, C(x) being, for every x ∈ &UOmega;, a bounded convex subset of R-N. The relaxed functionals G (1)(PC)((&UOmega;)), and G (W1,∞(&UOmega;)) of G obtained letting u vary in PC1 (&UOmega;), the set of the piecewise C-1-functions in &UOmega;, and in W-1,W-∞(&UOmega;) respectively in the definition of G are considered. Identity and integral representation results are proved under continuity-type assumptions on C, together with the description of the common density by means of convexification arguments. Classical relaxation results are extended to the case of the continuous variable dependence of C, and the non-identity features described in the measurable dependence case by De Arcangelis, Monsurro and Zappale (2004) are shown to be non-occurring. Proofs are based on the properties of certain limits of multifunctions, and on an approximation result for functions u in W-1,W-∞(&UOmega;), with &DEL; u(x) ∈ C(x) for a.e. x ∈ &UOmega;, by PC1 (&UOmega;) ones satisfying the same condition. Results in more general settings are also obtained.