Abstract:
The thermal equilibrium state of a bipolar, isothermic quantum fluid confined to a bounded domain ,d = 1,2 or d = 3 is entirely described by the particle densities n, p, minimizing the energy
where G 1,2 are strictly convex real valued functions, . It is shown that this variational problem has a unique minimizer in
and some regularity results are proven. The semi-classical limit is carried out recovering the minimizer of the limiting functional. The subsequent zero space charge limit leads to extensions of the classical boundary conditions. Due to the lack of regularity the asymptotics can not be settled on Sobolev embedding arguments. The limit is carried out by means of a compactness-by-convexity principle.
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Received: 26 March 1996 / Accepted: 13 January 1997
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Unterreiter, A. The Thermal Equilibrium Solution of a Generic Bipolar Quantum Hydrodynamic Model . Comm Math Phys 188, 69–88 (1997). https://doi.org/10.1007/s002200050157
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DOI: https://doi.org/10.1007/s002200050157