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A variational analysis of the thermal equilibrium state of charged quantum fluids. (English) Zbl 0820.35112

Summary: The thermal equilibrium state of a charged, isotropic quantum fluid in a bounded domain \(\Omega\) is entirely described by the particle density \(n\) minimizing the total energy \[ E(n) = \int_ \Omega \bigl | \nabla \sqrt n \bigr |^ 2 + \int_ \Omega H(n) + {1 \over 2} \int_ \Omega n V[n] + \int_ \Omega V^ en \] where \(\Phi = V[n] + V^ e\) solves Poisson’s equation \(- \Delta \Phi = n - C\) 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 \[ \biggl \{n \in L^ 1 (\Omega) : n \geq 0, \int_ \Omega n = N,\;\sqrt n \in H^ 1(\Omega) \biggr\}. \] Furthermore it is proven that \(n \in C^{1, \lambda}_{\text{loc}} (\Omega) \cap L^ \infty (\Omega)\) for all \(\lambda \in (0,1)\) and \(n > 0\) in \(\Omega\).

MSC:

35Q35 PDEs in connection with fluid mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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