부산대학교 도서관

We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as s→1Γε1-s<<ε2εε→0s→1. In a seminal paper by Bourgain et al. (Another look at Sobolev spaces. In: Optimal control and partial differential equations. IOS, Amsterdam, pp 439–455, 2001) it is proven that if the coefficient is constant then this sequence s→1Γε1-s<<ε2εε→0s→1-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by s→1Γε1-s<<ε2εε→0s→1 the scale of the oscillations and we assume that s→1Γε1-s<<ε2εε→0s→1, this sequence converges to the homogenized functional formally obtained by separating the effects of s and s→1Γε1-s<<ε2εε→0s→1; that is, by the homogenization as s→1Γε1-s<<ε2εε→0s→1 of the Dirichlet integral with oscillating coefficient obtained by formally letting s→1Γε1-s<<ε2εε→0s→1 first.