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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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