Relaxation and gamma-convergence of supremal functionals

Francesca Prinari

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 1, page 101-132
  • ISSN: 0392-4033

Abstract

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We prove that the Γ -limit in L μ of a sequence of supremal functionals of the form F k ( u ) = μ - ess sup Ω f k ( x , u ) is itself a supremal functional. We show by a counterexample that, in general, the function which represents the Γ -lim F ( , B ) of a sequence of functionals F k ( u , B ) = μ - ess sup B f k ( x , u ) can depend on the set B and wegive a necessary and sufficient condition to represent F in the supremal form F ( u , B ) = μ - ess sup B f ( x , u ) . As a corollary, if f represents a supremal functional, then the level convex envelope of f represents its weak* lower semicontinuous envelope.

How to cite

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Prinari, Francesca. "Relaxation and gamma-convergence of supremal functionals." Bollettino dell'Unione Matematica Italiana 9-B.1 (2006): 101-132. <http://eudml.org/doc/289598>.

@article{Prinari2006,
abstract = {We prove that the $\Gamma$-limit in $L^\infty_\mu$ of a sequence of supremal functionals of the form $F_k (u)=\operatorname\{\mu-ess\,sup\}_\Omega f_k(x, u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\Gamma$-lim $F(\cdot, B)$ of a sequence of functionals $F_k(u, B)= \operatorname\{\mu-ess\,sup\}_B f_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u, B)= \operatorname\{\mu-ess\,sup\}_B f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope.},
author = {Prinari, Francesca},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {101-132},
publisher = {Unione Matematica Italiana},
title = {Relaxation and gamma-convergence of supremal functionals},
url = {http://eudml.org/doc/289598},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Prinari, Francesca
TI - Relaxation and gamma-convergence of supremal functionals
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/2//
PB - Unione Matematica Italiana
VL - 9-B
IS - 1
SP - 101
EP - 132
AB - We prove that the $\Gamma$-limit in $L^\infty_\mu$ of a sequence of supremal functionals of the form $F_k (u)=\operatorname{\mu-ess\,sup}_\Omega f_k(x, u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\Gamma$-lim $F(\cdot, B)$ of a sequence of functionals $F_k(u, B)= \operatorname{\mu-ess\,sup}_B f_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u, B)= \operatorname{\mu-ess\,sup}_B f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope.
LA - eng
UR - http://eudml.org/doc/289598
ER -

References

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