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Journal of Convex Analysis 10 (2003), No. 1, 063--088
Copyright Heldermann Verlag 2003

Minimizers of Energy Functional under not very Integrable Constraints

Christian Léonard
Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France

We consider a general class of problems of minimization of convex integral functionals (such as entropy maximization) subject to linear constraints. Under general assumptions, the minimizers are characterized. Our results improve previous literature on the subject in the following directions:
-- necessary and sufficient conditions for the shape of the minimizers are proved
-- without constraint qualification
-- under infinitely many linear constraints subject to natural integrability conditions (no topological restrictions).
This paper extends previous results of the author by relaxing some integrability conditions on the constraint. As a consequence, the minimizers may admit a singular component. Our proofs mainly rely on convex duality.

Keywords: Maximum entropy method, relative entropy, convex integral functionals, convex conjugacy.

MSC 2000: 49K22, 52A41, 46B10.

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