Semicontinuity of an integral functional in Banach space
Copyright policy )Semicontinuity of an integral functional in Banach space
In the paper under review the author studies lower semicontinuity of an integral functional \(J_f :(x, u) \to \int_T f(x(t),u(t))dt\) that arises in various minimization problems. It is assumed that \(T\) equals \([0,1]\) and \(x\), \(u\) are elements in the space \(L_1(T,X)\) of all Bochner integrable functions taking values in a separable Banach space \(X\). Let a function \(f: X\times Y\to {\mathbb R}^+\cup\{{\infty}\}\) satisfy the following conditions: (a) the mapping \(x\to f^*(x,x^*)\) is upper semicontinuous for all \(x^*\) in the dual space \(X^*\) (\(f^*\) is the Fenchel-Moreau conjugate to f); (b) the mapping \(u\to f(x,u)\) is convex and lower semicontinuous. The author calls a sequence \(\{u_n\}\subset L_1(T,X)\) \(a\)-weakly convergent to \(u\in L_1(T,X)\) if it is bounded in \(L_1(T,X)\) and \(\int_E\langle x^*,u_n(t)\rangle \to \int_E\langle x^*,u(t)\rangle\) for all \(x^*\) in \(X^*\) and for all measurable subsets \(E\) of \([0,1]\). The main theorem is as follows. If \(x_n\) strongly converges to \(x\) and \(u_n\) \(a\)-weakly converges to \(u\), then \(J_f(x,u)\leq \liminf\limits_{n\to\infty} J_f(x_n,u_n)\). In this case, the functional \(J_f\) is called sequentially strong-\(a\)-weak lower semicontinuous. Sequential lower semicontinuity of \(J_f\) in the classical strong-weak definition is implied by the author's theorem (see, for example, \textit{C. Olech} [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 25, 135-142 (1977; Zbl 0395.46026)]; \textit{G. Bottaro} and \textit{P. Oppezzi} [Boll. Unione Mat. Ital., V. Ser. B 17, 1290-1307 (1980; Zbl 0451.46029)]; \textit{C. Castaing} and \textit{P. Clauzure} [Acta Math. Vietnam. 7, No. 2, 139-170 (1984; Zbl 0557.49005)]).
lower semicontinuity, Methods involving semicontinuity and convergence; relaxation, sequential lower semicontinuity, integrand, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), integral functional
lower semicontinuity, Methods involving semicontinuity and convergence; relaxation, sequential lower semicontinuity, integrand, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), integral functional
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