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Quasiconvexity

  • Filip RindlerEmail author
Chapter
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Part of the Universitext book series (UTX)

Abstract

We saw in the Tonelli–Serrin Theorem 2.6 that convexity of the integrand (in the gradient variable) implies the weak lower semicontinuity of the corresponding integral functional. Moreover, we proved in Proposition 2.9 that if \(d = 1\) or \(m = 1\), then convexity of the integrand is also necessary for weak lower semicontinuity. In the vectorial case (\(d, m > 1\)), however, it turns out that one can find weakly lower semicontinuous integral functionals whose integrands are non-convex.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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