Stable solutions of differential inclusions and the relaxation theorem

Lunedi' 5 Novembre 2012 - Mikhail Sychev


Lunedi' 5 novembre 2012 alle ore 15:00 in aula 2AB45, Mikhail Sychev (Sobolev Institute of Mathematics, Novosibirsk, Russia) terra' un seminario dal titolo "Stable solutions of differential inclusions and the relaxation theorem"

We show that the existence theory for differential inclusions is elementary and follows from the standard facts of Functional Analysis and Calculus of Variations. In particular the "ideas" of Convex Integration are not required. A more interesting issue is the topic of stable solutions. We show that the functional which measures maximal oscillations produced by admissible functions weakly converging to a given one admits a number of fine properties and these observations allow to obtain new results in the Calculus of Variations. For any integral functional with extended-valued Caratheodory integrand we show that the set where the functional is both lower semicontinuous and stable is dense in the area of definition of the functional. The values the functional admits in this set completely define the lower semicontinuous extension (which is abstact in the general case) and the extended functional is also stable in this set.

Rif. int. M. Bardi, C. Marchi, G. Treu

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