Minimizers of energy functionals

We consider a general class of problems of minimization of convex integralfunctionals (maximization of entropy) subject to linear constraints. Undergeneral assumptions, the minimizing solutions are characterized. Our resultsimprove previous literature on the subject in the following directions: -anecessary and suficient condition for the shape of the minimizing densityis proved -without constraint qualification -under infinitely many linearconstraints subject to natural integrability conditions (no topological restrictions).As an illustration, we give the general shape of the minimizing density forthe marginal problem on a product space. Finally, a counterexample of I. Csiszáris clarified. Our proofs mainly rely on convex duality.