A variational analysis of the thermal equilibrium state of charged quantum fluids

The thermal equilibrium state of a charged, isentropic quantum fluid in a bounded domain Ω is entirely described by the particle density n minimizing the total energy where π = V[n] + V e solves Poisson's equation -δπ = n-C subject to mixed Dirichlet-Neumann boundary conditions. It is shown that for given N > 0 (i.e. for prescribed total number of particles) this energy functional admits a unique minimizer in Furthermore it is proven that for all λ ∈ (0,1) and n>0 in Ω.