1,721,038 research outputs found
GLOBAL BIFURCATION ANALYSIS OF MEAN FIELD EQUATIONS AND THE ONSAGER MICROCANONICAL DESCRIPTION OF TWO-DIMENSIONAL TURBULENCE
No full textOn strictly starshaped domains of second kind (see Definition 1.2) we find sufficient
conditions which allow the solution of two long standing open problems closely related to the
mean field equation (P_\lambda) below.
On one side we describe the global behaviour of the Entropy for the mean field Microcanonical
Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional
turbulence. This is the completion of well known results first established in [12]. Among other
things we find a full unbounded interval of strict convexity of the Entropy.
On the other side, to achieve this goal, we have to provide a detailed qualitative description
of the global branch of solutions of (P) emanating from \lambda = 0 and crossing \lambda = 8\pi. This
is the completion of well known results first established in [32] and [14] for 8, and it
has an independent mathematical interest, since the shape of global branches of semilinear
elliptic equations, with very few well known exceptions, are poorly understood. It turns out
that the (MVP) suggests the right variable (which is the energy) to be used to obtain a global
parametrization of solutions of (P_\lambda). A crucial spectral simplification is obtained by using the
fact that, by definition, solutions of the (MVP) maximize the entropy at fixed energy and total
vorticity
Nonlinear elliptical problems with singular data and applications to Glashow-Salam-Weinberg electroweak theory
No full textOn the uniqueness and monotonicity of solutions of free boundary problems
No full textFor any smooth and bounded domain Ω⊂RN, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on Ω in a neat interval depending only by the best constant of the Sobolev embedding H01(Ω)↪L2p(Ω), [Formula presented] and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for p>1, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for p=1. The threshold, which is sharp for p=1, yields a new condition which guarantees that there is no free boundary inside Ω. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes
Stable and unstable equilibria of uniformly rotating self-gravitating cylinders
No full textThe equilibrium configurations of self-gravitating uniformly rotating isothermal cylinders
in contact with a heat bath and their stability is studied by recently derived analytical
techniques. The known critical temperature T c obtained by Katz and Lynden-Bell is
found to be a stability threshold with respect to axially symmetric perturbations. We
provide the almost explicit expression of negative specific heat solutions whose densities
are sharply concentrated either near the symmetry axis or near some off-axis filaments
as T → T c − . The critical angular frequency observed numerically in literature is found
to be the threshold value for the existence of these off-axis filaments. This is in strik-
ing contrast with the static case analyzed by Katz and Lynden-Bell where equilibrium
configurations are found only if T > T c and no negative specific heat equilibria exists at
all. Metastability of the free energy’s relative maximizers for T ≤ T c is also discussed.
Those off-axis configurations were predicted in the study of negative temperature states
for guiding-centre plasmas and vortex systems
A compactness result for periodic multivortices in the Electroweak Theory
No full textWe derive a priori uniform bounds for solutions of an elliptic system of Liouville-type equations, first analyzed by J. Spruck and Y. Yang (Comm. Math. Phys. 144 (1992) 1), yielding periodic multivortices in the classical electroweak theory of Glashow-Salam-Weinberg. Our proof is based on a concentration-quantization result, in the same spirit of Brezis-Merle (Comm. Partial Differential Equations 16 (8,9) (1991) 1223) and Li-Shafrir (Indiana Univ. Math. J. 43 (4) . (1994) 1255), for mean field equations on Riemannian compact 2-manifolds
On the best pinching constant of conformal metrics on S^2 with one and two conical singularities
No full textWe answer a long-standing open question asked by Thurston (The Geometry and Topology of Three-Manifolds. Princeton University Press, Princeton, 1978) concerning the best pinching constant for conformal metrics on S2 with one and two conical singularities of angles 2π(1+α 1) and 2π(1+α 1),2π(1+α 2) in case α 1∈(−1,0) and −1<α 1<α 2<0, respectively. The case of one conical singularity is a corollary of a result in Chen and Lin (Commun. Anal. Geom. 6(1):1–19, 1998) concerning the curvature of conformal metrics on ℝ2 with bounded Gaussian curvature 0<a≤K≤b<+∞. The case with two conical singularities is worked out by a generalization of that result
A "Sup + C Inf" inequality for the equation "
No full textWe generalize the ‘sup + C inf’ inequality obtained by Shafrir to the solutions of
−∆u =Ve u|x| 2αin Ω,
with Ω ⊂ R 2 open and bounded, α ∈ (0, 1) and V any measurable function which
satisfies 0 < a V
b < +∞
A Sup + Inf inequality for Liouville type equations with weights
No full textWe generalize a result by H. Brezis, Y. Y. Li and I. Shafrir [6]
and obtain an Harnack type inequality for solutions of − u = |x| 2α Ve u in for
⊂ R 2 open, α ∈ (−1, 0) and V any Lipschitz continuous function satisfying
0 < a ≤ V ≤ b < ∞ and ∇V ∞ ≤ A
A sup x inf-type inequality for conformal metrics on Riemann surfaces with conical singularities
No full textA sup × inf-type inequality is proved for the regular part of conformal factors for Rieman-
nian metrics on surfaces with conical singularities of positive order α > 0. The proof is
based on the explicit representation formula for solutions of the singular Liouville equa-
tion and generalizes to the singular case an argument by I. Shafrir
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