An Introduction to Γ-Convergence by Gianni Dal Maso (auth.)

By Gianni Dal Maso (auth.)

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Ltoc(n)) and such that lim H( Uh) exists and is Proof. h-+oo less than +00. By (a) the sequence (Uh) is bounded in WI,p(n), and, since WI,p(n) is reflexive (recall that p > 1), a subsequence of (Uh) converges to a function v in the weak topology of WI,p(n). Since (Uh) converges to u in the strong topology of LP(n) (resp. Ltoc(n)), we have u = v, hence u E WI,p(n) and (Uh) converges to u in the weak topology of WI,p(n). The conclusion H(u):5 lim H(Uh) follows now from the lower semicontinuity of H in the h-+oo weak topology of WI,p(n).

10. The K-lower limit of the sequence (Eh), denoted by K-lim inf Eh, is the set of all points x E X with the following property: h-+oo for every U E N{x) there exists kEN such that Un Eh 1: 0 for every h ~ k. The K-upper limit, denoted by K-lim sup Eh, is the set of all points h-+oo 42 An Introduction to r -convergence x E X with the following property: for every U E N(x) and for every kEN there exists h ~ k such that un Eh =1= 0. If there exists a set E ~ X such that E = K-lim inf Eh = K-lim sup Eh then we write E = K- lim Eh, and h-+oo h-+oo h-+oo we say that the sequence (Eh) converges to E in the sense of Kuratowski, or K-converges to E (in X).

14. Let X K-converges to {O}. and let Eh 43 = [0, Ilh] U [h, +00[. 6. It is defined by XE(x) = 0, if x E E, and XE(x) = +00, if x E X \ E. The following proposition shows that the K-convergence of a sequence of sets is equivalent to the r -convergence of the corresponding indicator functions. 15. Let (Eh) be a sequence of subsets of X, and let E' = K-liminf Eh, E" = K-limsupEh. h~oo h~oo Then XE" = r -lim inf XEh . h~oo In particular (Eh) K-converges to E in X if and only if (XEJ r-converges to XE in X.

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