Journal of Convex Analysis Volume 9 (2002), No. 1, 225–236 The Class of Functionals which can be Represented by a Supremum Emilio Acerbi Dipartimento di Matematica, Universit`a di Parma, Via D’Azeglio 85/A, 43100 Parma, Italy acerbi@prmat.math.unipr.it Giuseppe Buttazzo Dipartimento di Matematica, Universit`a di Pisa, Via Buonarroti 2, 56127 Pisa, Italy buttazzo@dm.unipi.it Francesca Prinari Dipartimento di Matematica, Universit`a di Pisa, Via Buonarroti 2, 56127 Pisa, Italy prinari@mail.dm.unipi.it Received January 22, 2001 Revised manuscript received March 29, 2001 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. Keywords: Performance function, multipliers, stability, convex like functions, measurable integrands, richness, integral functional, growth conditions 1991 Mathematics Subject Classification: 46E30, 28A20, 49B, 60B12 1. Introduction Functionals which can be written in supremal form F (u, B)= μ - sup  f ( x, u(x) ) : x ∈ B  (1) received much attention in the last years (see References). In the applications they de- scribe optimization problems whose criteria select solutions which minimize a given quan- tity 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) is weakly* L ∞ sequentially lower semicontinuous if and only if the function ISSN 0944-6532 / $ 2.50 c  Heldermann Verlag