A fundamental aspect of the study of N−electronic systems (systems containing N electrons) is to obtain information on the states in which these systems have minimal energy. In practice a numerical search of such states is impossible to carry out, so that alternative approaches have been developped, the one around which this work revolves being to consider electronic systems through their electronic density rather than their state. This approach, known today as Density Functional Theory (DFT), was formalised in Kohn and Sham’s seminal article  and its mathematical aspects were studied a few years later by Lieb . Since then, the ideas leading to the construction of DFT have been adapted to the context of electronic systems with a fractionnal number of electrons (open systems), first through PPLB DFT and more recently through the definition of N−centered DFT[4, 5]. In both cases it is unclear wherether the mathematical properties established for classical DFT can be expected to hold true. This question is the main problematic of our work, in which we shall study the analogy between N−centered and classical DFT, from their construction to the methods that are derived from them. This will lead us to construct a Kohn-Sham scheme for N−centered DFT, investigate the links between this theory and optimal transport and present the Hubbard Dimer in this particular situation.