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Interpretation of Lavrentiev phenomenon by relaxation: the higher order case


Author: Marino Belloni
Journal: Trans. Amer. Math. Soc. 347 (1995), 2011-2023
MSC: Primary 49J45; Secondary 49J05
DOI: https://doi.org/10.1090/S0002-9947-1995-1290714-9
MathSciNet review: 1290714
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Abstract: We consider integral functionals of the calculus of variations of the form

$\displaystyle F(u) = \int\limits_0^1 {f(x,u,u', \ldots ,{u^{(n)}})dx} $

defined for $ u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $ F$ with respect to the weak $ W_{{\text{loc}}}^{n,1}(0,1)$ convergence can be written as

$\displaystyle \overline F (u) = \int\limits_0^1 {f(x,u,u', \ldots ,{u^{(n)}})dx + L(u),} $

where the additional term $ L(u)$, the Lavrentiev Gap, is explicitly identified in terms of $ F$.

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DOI: https://doi.org/10.1090/S0002-9947-1995-1290714-9
Article copyright: © Copyright 1995 American Mathematical Society

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