ISSN:
1573-0530
Keywords:
35Q53
;
53A10
;
76C05
;
82D50
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract Ifλ ⩾ 0 and Λ ⊂ ℝ2 is a connected Lipschitz domain, with |Λ| 〈 ∞, it is a matter of standard convex functional analysis to show that the nonlinear eigenvalue problem of Poisson-Boltzmann type $$ - \Delta \psi (x) = {{\lambda \sum\limits_{z = \pm 1} z } \mathord{\left/ {\vphantom {{\lambda \sum\limits_{z = \pm 1} z } {\int_\Lambda {\exp (z[\psi (x) - \psi (y)])d^2 y,} }}} \right. \kern-\nulldelimiterspace} {\int_\Lambda {\exp (z[\psi (x) - \psi (y)])d^2 y,} }}$$ with 0-Dirichlet boundary data forψ, has a unique solutionψ 0 Ξ 0. Here, we prove the stronger result thatψ =ψ 0 Ξ 0 is the unique solution also forλ ∈ (λ *, 0), whereλ * 〈 0 is some critical value which depends only on Λ, but in any event withλ * 〈 -8π/5. This result settles a conjecture about negative temperatures of vorticity compounds in 2D turbulence which goes back to 1949 work of Onsager.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00739374
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