A note on the Poisson-Boltzmann equation

Steady States for Streater’s Energy-Transport Models of Self-Gravitating Particles

“…We recall an argument based on the Pohozaev method in the proofs of Remark 5.1 and Theorem 5.7. In the two-dimensional case this nonexistence result has been extended to an arbitrary simply connected domain, see [24].…”

Asymptotic Behavior of an Initial-Boundary Value Problem for the Vlasov--Poisson--Fokker--Planck System

“…Lions [26] showed the existence of a unique regular solution of problem (10)- (11) in the electrostatic case in the space H 1 o (Ω). In the gravitational case the PB equation has been studied by A. Krzywicki and T. Nadzieja in [28,29]. They proved that problem (10)-(11) has no classical solutions for α > 0 large enough for simply connected domains.…”

“…) independently of n. Applying the equation of the nonlinear change of variables, (23) of Lemma 3.9, to ϕ (f n ), we obtain (29) in the sense of the distributions on [0, T ] × Ω × R 3 . As…”

“…u Ω e u dx (35) with Dirichlet boundary conditions, where α = β σ M . Equation (35) has solution(s) depending strongly on the topology of the domain Ω (see [28,29]). In fact, if Ω is the unit ball B(0, 1), we have the following properties:…”

Boundary Value Problems for the Vlasov-Maxwell System

“…In the case n = 1, boundary value problem (2.55), (2.56) was considered in Gogny, Lions [29], Krzywicki, Nadzieja [41]. 4.…”