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Relaxation and gamma-convergence of supremal functionals

Francesca Prinari — 2006

Bollettino dell'Unione Matematica Italiana

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.

Γ -convergence and absolute minimizers for supremal functionals

Thierry ChampionLuigi De PascaleFrancesca Prinari — 2004

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the L p approximants naturally associated to a supremal functional Γ -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local...

-convergence and absolute minimizers for supremal functionals

Thierry ChampionLuigi De PascaleFrancesca Prinari — 2010

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the approximants naturally associated to a supremal functional -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer ( local solution)...

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