Lieb functional domains

The functionals FHK and FEHK have the unfortunate mathematical difficulty that their domains of definition A and B, although they are well defined, are difficult to characterize, i.e., it is difficult to know if a given density n belongs to A or B. It is therefore desirable to extend the domains of definition of FHK and FEHK to an easily characterizable (preferably convex) set of densities. This can be achieved using the constrained search procedure introduced by Levy [19]. We define the Levy-Lieb functional FLL as ... 

As a next step we will show that this implies that the functional FL is Gateaux differentiable at every E-V-density and nowhere else. For the application of the next theorem it is desirable to extend the domain of FL to all of L1 DL3. We follow Lieb 1 and define... 

As mentioned, in order to be able to apply the variational principle in DFT, it is necessary to extend the definition of the functionals beyond the domain of v-representable densities, and the standard procedure is here to apply the Levy constrained-search procedure [17]. This has led to the functionals known as the Levy-Lieb (FL[p ) and Lieb (FL[p ) functionals, respectively, and we shall now investigate the differentiability of these functionals. This will represent the main part of our paper. 

Above we have assumed that the minimization is carried out within the domain of normalized of densities. Alternatively, we can perform the minimization, using the Euler-Lagrange procedure. Then we use the extension of the functionals valid also outside the normalization domain and enforce the normalization constraint by a Lagrange multiplier.5 For the Levy-Lieb energy functional (70) this leads to... 

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