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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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