Approximation of Usual and of Generalized Solutions
In this chapter we cover first the question of approximating absolute continuous solutions by means of C1 solutions for Lipschitzian problems (Section 18.1). In particular we state and proved Angell’s theorem, which includes most previous results, in terms of the same property (D) we have used in Chapter 13 for a far different purpose. In Section 18.2 we then present a simple example due to Manià which shows that the just-mentioned approximation, in general, may not be possible (Lavrentiev’s phenomenon). In Section 18.3 we present a proof that generalized solutions for Lipschitzian problems can be approximated by means of usual absolutely continuous solutions, and in Section 18.4 we show by means of an example that such an approximation, in general, may not be possible.
KeywordsGeneralize Solution Bibliographical Note Usual Solution Lagrange Problem Lavrentiev Phenomenon
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