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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Lower semicontinuity of integral functionals


Author: Leonard D. Berkovitz
Journal: Trans. Amer. Math. Soc. 192 (1974), 51-57
MSC: Primary 49A50
DOI: https://doi.org/10.1090/S0002-9947-1974-0348582-1
MathSciNet review: 0348582
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Abstract: It is shown that the integral functional $ I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu $ is lower semicontinuous on its domain with respect to the joint strong convergence of $ {y_k} \to y$ in $ {L_p}(G)$ and the weak convergence of $ {z_k} \to z$ in $ {L_p}(G)$, where $ 1 \leq p \leq \infty $ and $ 1 \leq q \leq \infty $, under the following conditions. The function $ f:(t,x,w) \to f(t,x,w)$ is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).


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DOI: https://doi.org/10.1090/S0002-9947-1974-0348582-1
Keywords: Lower semicontinuity, optimal control of distributed parameter systems, existence theorems in variational problems
Article copyright: © Copyright 1974 American Mathematical Society