Few remarks on differential inclusions
Mikhail A. Sytchev
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Submission date: 03. Sep. 2001
published in: Proceedings of the Royal Society of Edinburgh / A, 136 (2006) 3, p. 649-668
DOI number (of the published article): 10.1017/S0308210500005102
Keywords and phrases: differential inclusions, baire categories, convex integration, integral functionals, weak-strong convergence
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In this paper we analyse the methodology of the theory of differential inclusions. First we emphasize that any sequence of piece-wise affine functions with successive elements obtained by perturbations of preceding ones in the sets of their affinity converges strongly. This gives a simple algorithm to construct sequences of approximate solutions which converge to exact ones (neither specific choice suggested by the method of convex integration nor Baire category methodology is required). Then we suggest a functional which is defined in the set of admissible functions and which measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given one. This functional can be used to prove that the set of stable solutions is dense (residual) in the closure of the set of admissible functions both via the Baire category lemma or via specific choice of strictly convergent sequences.