Application of Young measures to uniformly integrable sequences in a separable Banach space. (Application des mesures de Young aux suites uniformément integrables dans un Banach séparable.) (French) Zbl 0742.49010

Valadier, Michel

Sémin. Anal. Convexe, Univ. Sci. Tech. Languedoc 20, Exp. No. 3, 14 p. (1990).

The main results of the paper extend to a separable Banach space \(E\) well-known theorems of functional analysis and the calculus of variations. More precisely, the author proves the weak-compactness of a sequence \((u_ n)_ n\) of uniformly integrable functions in \(L^ 1(\Omega,\mu;E)\) and a lower semicontinuity result for integral functionals of the type \[ I[u,v]=\int_ \Omega \psi(\omega,u(\omega),v(\omega))d\mu \] with respect to the convergence in measure of a sequence \((u_ n)_ n\) of measurable functions and the weak convergence of a sequence \((v_ n)_ n\) in \(L^ 1(\Omega,\mu;E)\). The technique used makes use of the theory of Young measures [see also the author in: “Methods of nonconvex analysis”, Lect. Notes Math. 1446, 152-188 (1990; Zbl 0738.28004)].

Reviewer: C.Vinti (Perugia)

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

separable Banach space; weak-compactness; uniformly integrable functions; Young measures

Citations:

Zbl 0738.28004