Abstract
Motivated by a recent interest in models of peridynamics, we survey, gather, and summarize various results, that are scattered throughout the literature, about weak lower semicontinuity of some non-local variational problems in which non-locality is expressed in the form of a double integral. Weak lower semicontinuity translates into separate convexity, though one has to incorporate in such a result a most unusual phrase to stress that weak lower semicontinuity ought to be true for all domains. The most striking fact, however, is that relaxation cannot be written and understood as a functional of the same kind but something more involved needs to be found.
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Supported by MINECO/FEDER grant MTM2013-47053-P, by PEII-2014-010-P of the Conserjería de Cultura (JCCM), and by grant GI20152919 of UCLM.
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Pedregal, P. Weak lower semicontinuity and relaxation for a class of non-local functionals. Rev Mat Complut 29, 485–495 (2016). https://doi.org/10.1007/s13163-016-0201-6
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DOI: https://doi.org/10.1007/s13163-016-0201-6