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Semicontinuity and Supremal Representation in the Calculus of Variations

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Abstract

We study the weak* lower semicontinuity properties of functionals of the form

$$F(u)=\mathop{\mathrm{ess\,sup}}_{x\in\Omega}f(x,Du(x))$$

where Ω is a bounded open set of R N and u∈W 1,∞(Ω). Without a continuity assumption on f(⋅,ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if F is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in general it is not possible to represent F through the level convex envelope of f.

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  1. Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, Via Prov. le Lecce-Arnesano, 73100, Lecce, Italy

    Francesca Prinari

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  1. Francesca Prinari
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Correspondence to Francesca Prinari.

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Prinari, F. Semicontinuity and Supremal Representation in the Calculus of Variations. Appl Math Optim 58, 111–145 (2008). https://doi.org/10.1007/s00245-007-9033-6

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  • DOI: https://doi.org/10.1007/s00245-007-9033-6

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