# Semicontinuity and Supremal Representation in the Calculus of Variations

Semicontinuity and Supremal Representation in the Calculus of Variations 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Semicontinuity and Supremal Representation in the Calculus of Variations

, Volume 58 (1) – Aug 1, 2008
35 pages

/lp/springer_journal/semicontinuity-and-supremal-representation-in-the-calculus-of-B7GbWksC2m
Publisher
Springer-Verlag
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-007-9033-6
Publisher site
See Article on Publisher Site

### 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 .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2008

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