131,341 research outputs found
On the classification of N-point concentrating solutions for mean field equations and the critical set of the N-vortex singular Hamiltonian on the unit disk
No full textMotivated by the analysis of the multiple bubbling phenomenon (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004) for a singular mean field equation on the unit disk (Bartolucci and Montefusco in Nonlinearity 19:611–631, 2006), for any N≥3 we characterize a subset of the 2π/N-symmetric part of the critical set of the N-vortex singular Hamiltonian. In particular we prove that this critical subset is of saddle type. As a consequence of our result, and motivated by a recently posed open problem (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004), we can prove the existence of a multiple bubbling sequence of solutions for the singular mean field equation
Uniqueness results for mean field equations with singular data
No full textWe prove uniqueness of solutions for mean field equations (Caglioti et. Al. Comm. Math. Phys. 174 (1995)) with singular data (Bartolucci et Al. Comm. Math. Phys. 229 (2002)),
arising in the analysis of two-dimensional turbulent Euler flows.
In this way, we generalize to the singular case some uniqueness results obtained by
Chang, Chen and the second author (Chang et. Al. New Stud. Adv. Math. 2 (2003)). In particular,
by using a sharp form of an improved
Alexandrov-Bol's type isoperimetric inequality, we are able to exploit the role played by
the singularities and then obtain uniqueness under weaker boundary regularity assumptions than those
assumed in (Chang et. Al. New Stud. Adv. Math. 2 (2003))
n the Ambjorn-Olesen electroweak condensates
No full textWe obtain sufficient conditions for the existence of the Ambjorn-Olesen [“On elec-
troweak magnetism,” Nucl. Phys. B315, 606–614 (1989)] electroweak N-vortices
in case N ≥ 1 and therefore generalize earlier results [D. Bartolucci and G. Taran-
tello, “Liouville type equations with singular data and their applications to periodic
multivortices for the electroweak theory,” Commun. Math. Phys. 229, 3–47 (2002);
J. Spruck and Y. Yang, “On multivortices in the electroweak theory I: Existence of
periodic solutions,” ibid. 144, 1–16 (1992)] which handled the cases N ∈ {1, 2, 3,
4}. The variational argument provided here has its own independent interest as it
generalizes the one adopted by Ding et al. [“Existence results for mean field equa-
tions,” Ann. Inst. Henri Poincare, Anal. Non Lineaire 16, 653–666 (1999)] to obtain
solutions for Liouville-type equations on closed 2-manifolds. In fact, we obtain at
once a second proof of the existence of supercritical conformal metrics on surfaces
with conical singularities and prescribed Gaussian curvature recently established by
Bartolucci, De Marchis and Malchiodi [Int. Math. Res. Not. 24, 5625–5643 (2011)].
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4731239
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
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
The newborn hearing screening. Screening neonatale dell’udito. Indagine in Italia
No full textBroglio D., Bartolucci M. A., Bubbico L., The newborn hearing screening. Screening neonatale dell’udito. Indagine in Italia, in "Italian Journal of Pediatrics", 2005. Isfol OA: the newborn hearing screening. screening neonatale dell’udito. indagine in italia d. broglio m. a. bartolucci luciano bubbic
Nonlinear elliptical problems with singular data and applications to Glashow-Salam-Weinberg electroweak theory
No full textA 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
Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter
No full textWe consider the mean field equation:
Δu + ρ
u = 0
e u
e u
Ω
= 0 in Ω,
(1)
on ∂Ω,
where Ω ⊂ R 2 is an open and bounded domain of class C 1 . In his 1992 paper, Suzuki
proved that if Ω is a simply-connected domain, then Eq. (1) admits a unique solution
for ρ ∈ [0, 8π ). This result for Ω a simply-connected domain has been extended to the
case ρ = 8π by Chang, Chen and the second author. However, the uniqueness result
for Ω a multiply-connected domain has remained a long standing open problem which
we solve positively here for ρ ∈ [0, 8π ]. To obtain this result we need a new version
of the classical Bol’s inequality suitable to be applied on multiply-connected domains.
Our second main concern is the existence of solutions for (1) when ρ = 8π . We obtain
a necessary and sufficient condition for the solvability of the mean field equation at
ρ = 8π which is expressed in terms of the Robin’s function γ for Ω. For example,
if Eq. (1) has no solution at ρ = 8π , then γ has a unique nondegenerate maximum
point. As a by product of our results we solve the long-standing open problem of the equivalence of canonical and microcanonical ensembles in the Onsager’s statistical
description of two-dimensional turbulence on multiply-connected domains
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
- …
