Relaxation and convexity of functionals with pointwise nonlocality
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Abstract:
It is shown that the relaxation of the integral functional involving argument deviations \[ I(u):=\int _\Omega f(x,\{u_i(g_{ij}(x))\}_{i,j=1}^{k,l}) d\mu _\Omega (x), \] 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.References
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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
- Received by editor(s): June 15, 2000
- Published electronically: August 7, 2001
- Communicated by: Jonathan M. Borwein
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1862123