By Gianni Dal Maso (auth.)

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Extra resources for An Introduction to Γ-Convergence

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Note that in this case the r -limit and the pointwise limit are different at every point x E R. Let us return to a general topological space X. 5. , for every hEN there exists a constant ah E R such that Fh(X) = ah for every x EX, then (r-liminf Fh)(X) = liminf ah, h ..... oo h ..... oo (r-limsupFh)(x) = limsupah h ..... oo h-+oo for every x EX. , there exists F: X -+ R such that Fh(X) = F(x) for every x E X and for every hEN, then r-liminf Fh = r-limsup Fh = sc- F, h ..... oo h ..... 3). 6. Let N = N U {oo} with the usual compact topology, and let G: N x X ~ R be the function defined by G(h, x) = { Fh(X), if hEN, +00, if h = 00.

If (Fh) is an increasing sequence, then r- lim Fh h .... oo Proof. For every open set U = ~ lim sc- Fh h ..... oo = sup sc- Fh. hEN X we have lim inf Fh(Y) h-+oo yEU = sup inf Fh(Y) , hEN yEU hence for every x E X sup lim inf Fh(Y) UEN(x) h-+oo yEU = sup sup = sup sup inf Fh(Y) UEN(x) hEN yEU inf H(y) hEN UEN(x) yEU = sup (sc- Fh)(X) , hEN which concludes the proof of the proposition. 5. 4. The following example shows that this property does not hold if the functions Fh are not lower semicontinuous.

1). 3. If (Fh) converges to F uniformly and each function Fh is lower semicontinuous, then F is lower semicontinuous, hence (Fh) r -converges to F. 4. If (Fh) is an increasing sequence, then r- lim Fh h .... oo Proof. For every open set U = ~ lim sc- Fh h ..... oo = sup sc- Fh. hEN X we have lim inf Fh(Y) h-+oo yEU = sup inf Fh(Y) , hEN yEU hence for every x E X sup lim inf Fh(Y) UEN(x) h-+oo yEU = sup sup = sup sup inf Fh(Y) UEN(x) hEN yEU inf H(y) hEN UEN(x) yEU = sup (sc- Fh)(X) , hEN which concludes the proof of the proposition.

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