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The class of functionals which can be represented by a supremum

Summary: The class of functionals
which can be represented by a supremum
Emilio Acerbi Giuseppe Buttazzo Francesca Prinari
Abstract. We give a characterization of all lower semicontinuous functionals on L
µ which
can be represented in the form µ - sup{f(x, u): x A}. We also show by a counterexample that
the representation above may fail if the lower semicontinuity condition is dropped.
1. Introduction
Functionals which can be written in supremal form
(1.1) F(u, B) = µ - sup f x, u(x) : x B
received much attention in the last years (see References). In the applications they describe
optimization problems whose criteria select solutions which minimize a given quantity in the worst
possible situation. This is for instance the case of criteria like the maximum stress in elasticity,
the maximum loss in economy, the maximum pressure in problems from fluidodynamics.
In order to apply the direct methods of the calculus of variations to this class of functionals,
a first problem to be solved is the identification of qualitative conditions on the supremand f
which imply the lower semicontinuity with respect to a convergence weak enough to provide the
compactness in a large number of situations, say the weak* L
convergence. This was already
solved by Barron and Liu in [3] where they showed that a functional of the form (1.1) is weakly*


Source: Acerbi, Emilio - Dipartimento di Matematica, Universitą degli Studi di Parma


Collections: Mathematics