Abstract Methods in Partial Differential Equations by Carroll Robert W

Author:Carroll, Robert W.
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2013-06-05T16:00:00+00:00

Indeed for one has (E′ k, h) = (k, E′h) (exercise) and, since V(t) ⊂ K, clearly (u(t), E′(t)k) = (E′(t)u(t), k) a.e. Further, E′u ∈ L2(Ta, H), since, for any h ∈ H, (E′( · )u( · ), h) is measurable as a pointwise limit a.e. of a sequence of measurable functions (E′( · )u( · ), kn) = (u( · ), E′( · )kn), where kn → h, kn ∈ K (recall also that ||E′(t)| V(t)|| ≤ c). Using the density of K in H again and either observing that (cf. section 1.11), or simply extending (5.33) to any k ∈ H by continuity and applying equation (1.11.14), we conclude that

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