NEGATIVE TEMPERATURE BOUNDS FOR 2D VORTICITY COMPOUNDS

If lambda greater than or equal to 0 and Lambda subset of R(2) is a connected Lipschitz domain, with \Lambda\ < infinity, it is a matter of standard covex functional analysis to show that the nonlinear eigenvalue problem of Poisson-Boltzmann type [GRAPHICS] with 0-Dirichlet boundary data for psi, has a unique solution psi(0) = 0. Here, we prove the stronger result that psi = psi(0) = 0 is the unique solution also for lambda is an element of (lambda*, 0), where lambda* < 0 is some critical value which depends only on Lambda, but in any event with lambda* < -8 pi/5. This result settles a conjecture about negative temperatures of vorticity compounds in 2D turbulence which goes back to 1949 work of Onsager.