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Relaxation and convexity of functionals with pointwise nonlocality


Author: Eugene Stepanov
Journal: Proc. Amer. Math. Soc. 130 (2002), 433-442
MSC (2000): Primary 49J45; Secondary 47B37, 47H30, 49J25
DOI: https://doi.org/10.1090/S0002-9939-01-06281-5
Published electronically: August 7, 2001
MathSciNet review: 1862123
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Abstract | References | Similar Articles | Additional Information

Abstract:

It is shown that the relaxation of the integral functional involving argument deviations

\begin{displaymath}I(u):=\int_\Omega f(x,\{u_i(g_{ij}(x))\}_{i,j=1}^{k,l})\, d\mu_\Omega(x), \end{displaymath}

in weak topology of a Lebesgue space $(L^p(\Theta,\mu_\Theta))^k$ (where $(\Omega,\Sigma(\Omega),\mu_\Omega)$ and $(\Theta,\Sigma(\Theta),\mu_\Theta)$are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions $g_{ij}$: $\Omega\to \Theta$ satisfies the special condition, called unifiability, which can be regarded as collective nonergodicity or commensurability property, and is automatically satisfied only if $k=l=1$. If, however, either $k>1$ or $l>1$, then it is shown that as opposed to the classical case without argument deviations, for nonunifiable function matrix $\{g_{ij}\}$ one can always construct an integrand $f$ so that the functional $I$ itself is already weakly lower semicontinuous but not convex.


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Additional Information

Eugene Stepanov
Affiliation: Dipartimento di Matematica, Universitá di Pisa, via Buonarroti 2, 56127 Pisa, Italy
Email: stepanov@cibs.sns.it

DOI: https://doi.org/10.1090/S0002-9939-01-06281-5
Received by editor(s): June 15, 2000
Published electronically: August 7, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society

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