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In this paper, we prove that the Lp approximants naturally associated to a supremal functionalΓ-converge to it. This yields a lower semicontinuity result for supremalfunctionals whose supremand satisfy weak coercivity assumptions aswell as a generalized Jensen inequality. The existence of minimizersfor variational problems involving such functionals (together with aDirichlet condition) then easily follows. In the scalarcase we show the existence of at least one absolute minimizer (i.e. localsolution) among these minimizers. We provide two different proofs ofthis fact relying on different assumptions and techniques.

In this paper we study the lower semicontinuity problem for a supremalfunctional of the form $F(u,\Omega )= \underset{x\in\Omega}{\rm ess\,sup} f(x,u(x),Du(x))$ with respect to the strong convergence in L ∞(Ω), furnishing a comparison with the analogous theory developed bySerrin for integrals. A sort of Mazur's lemma for gradients of uniformlyconverging sequences is proved.

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Γ -convergence and absolute minimizers for supremal functionals

On the Lower Semicontinuity of Supremal Functionals

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Γ -convergence and absolute minimizers for supremal functionals

On the Lower Semicontinuity of Supremal Functionals