A Variational Analysis Of The Thermal-Equilibrium State Of Charged Quantum Fluids - Pacard, F.; Unterreiter, A. (Communications in Partial Differential Equations, 1995)
The thermal equilibrium state of a charged, isentropic quantum fluid in a bounded domain Omega is entirely described by the particle density n minimizing the total energy E(n) = integral(Omega)\backslashdel root n\backslash(2) + integral(Omega)H(n) + 1/2 integral(Omega)nV[n] + integral(Omega)V(e)n 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 {n is an element of L(1) (Omega); n greater than or equal to 0, integral(Omega) n = N, root n is an element of H-1 (Omega)} Furthermore it is proven that n is an element of C-loc(1,lambda)(Omega)boolean AND L(infinity)(Omega) for all lambda is an element of (0, 1) and n > 0 in Omega.
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