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Necessary and sufficient conditions for $$L_ 1$$-strong-weak lower semicontinuity of integral functionals. (English) Zbl 0638.49004
Functionals of the form $$F(u,v)=\int_{\Omega}f(x,u(x),v(x))d\mu (x)$$ are considered, where ($$\Omega$$,$${\mathcal F},\mu)$$ is a finite non-atomic measure space, $$f: \Omega\times U\times V\to]-\infty,+\infty]$$ is $${\mathcal F}\times {\mathcal B}(U\times V)$$-measurable U and V being separable Banach spaces with V reflexive, and $$u\in L$$ 1($$\Omega$$ ;U), $$v\in L$$ 1($$\Omega$$ ;V). The main result is the following:
The conditions
(i) f(x,$$\cdot,\cdot)$$ is sequentially l.s.c. on $$U\times V$$ for $$\mu$$- a.e. $$x\in \Omega;$$
(ii) f(x,u,$$\cdot)$$ is convex on V for $$\mu$$-a.e. $$x\in \Omega$$ and every $$u\in U;$$
(iii) there exist $$a\in L$$ 1($$\Omega)$$ and $$b>0$$ such that $$f(x,u,v)\geq a(x)-b(\| u\| +\| v\|)$$ for $$\mu$$-a.e. $$x\in \Omega$$, $$u\in U$$, $$v\in V$$ are sufficient for the sequential strong-weak lower semicontinuity of F on L 1($$\Omega$$ ;U)$$\times L$$ 1($$\Omega$$ ;V). Moreover, they are also necessary, provided that $$F(u_ 0,v_ 0)<+\infty$$ for some $$u_ 0\in L$$ 1($$\Omega$$,U) and $$v_ 0\in L$$ 1($$\Omega$$ ;V).
Reviewer: G.Buttazzo

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 46B99 Normed linear spaces and Banach spaces; Banach lattices 49J27 Existence theories for problems in abstract spaces
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##### References:
  Balder, E.J., On seminormality of integral functionals and their integrands, SIAM J. control optim., 24, 95-121, (1986) · Zbl 0595.49008  Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, () · Zbl 0346.46038  Cesari, L., Optimization—theory and applications, problems with ordinary differential equations, (1983), Springer Berlin · Zbl 0506.49001  Diestel, J.; Uhl, J.J., Vector measures, () · Zbl 0369.46039  Ioffe, A.D., On lower semicontinuity of integral functionals I, SIAM J. control optim., 15, 521-538, (1977) · Zbl 0361.46037  Olech, C., A characterization of L1-weak lower semicontinuity of integral functionals, Bull. acad. Pol. sci. Sér. sci. math. astr. phys., 25, 135-142, (1977) · Zbl 0395.46026  Poljak, B.T.; Poljak, B.T., Semicontinuity of integral functionals and existence theorems, Mat. sb., Math. U.S.S.R. sb., 7, 59-77, (1969) · Zbl 0164.42302  Rockafellar, R.T., Integrals which are convex functionals, Pacif. J. math., 24, 525-539, (1969) · Zbl 0159.43804  Tonelli, L., Sugli integrali del calcolo delle variazioni in forma ordinaria, Ann. scu. norm. sup. Pisa, 2, 3, 401-450, (1934) · JFM 60.0452.02  Nougues-Sainte-Beuve, M.F., Conditions nécessaires pour la semi-continuité inférieure d’une fonctionelle intégrale, Seminaire d’analyse convexe, université des sciences et techniques du languedoc, 15, 10.1-10.36, (1985)
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