From 1-homogeneous supremal functionals to difference quotients: relaxation and G-convergence

Garroni, Adriana; Ponsiglione, Marcello; Prinari, Francesca
December 2006
Calculus of Variations & Partial Differential Equations;Dec2006, Vol. 27 Issue 4, p397
Academic Journal
In this paper we consider positively 1-homogeneous supremal functionals of the type F(u) := supO f (x,?u(x)). We prove that the relaxation ... is a difference quotient, that is ...(u) = RdF (u) := sup x,y?,O x ?y u(x) - u(y)/dF (x, y) for every u ? W1,8(O), where dF is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to - convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances.


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