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Additional resources for Elements of the Differential and Integral Calculus
Example text
Hence, it suffices to show that each Fk is weakly lower semicontinuous, since the supremum of a family of lower-semicontinuous functionals is weakly lower semi continuous. Ik(U) dt :S Let Uj --" U in LI(a, b). l(b) we have J: J: Iduj) dt + J: Ik(u)(u - Uj) dt; that is, Fk(U) :S Fk(Uj) + lab I~(u)(u - Uj) dt. 5). Then, F is weakly lower semicontinuous in LP(a, b) if and only if I is lower semicontinuous and convex. 11. 5) on LI(a, b) is weakly lower semicontinuous in LI (a, b); hence its restriction to LP(a, b) is weakly lower semicontinuous in LP(a, b).
Since R is compact, there exists an increasing sequence of integers (aJ)j along which the limit exists, and for all k 2 1 we define (aj)j as any subsequence of (a;-l)j along which the limit r -convergence by numbers 36 exists. The 'diagonal' sequence jk = aZ, being a subsequence of all (a;)j, has the property that the limit exists for all lEN. 13). 42 fails. As an example, we can take X = {-I, l}N equipped with the discrete topology. X is metrizable, and f -convergence on X is equivalent to pointwise convergence.
On the other hand it can be checked (and is left as an exercise) that it also defines a convex and lower semicontinuous function which is less than f; hence also the converse inequality holds. 14) (just apply the direct methods). Note that this formula implies that if Zl and Z2 solve the minimum problem then j**(Zk) = f(Zk) for k = 1,2. This shows that j** coincides with g**, where g(z) = {f(Z) +00 if j**(z) = f(z) otherwise. 15) (d) The set J = {t** < +oo} is an interval on the closure of which j** is continuous.