Analytical Issues in the Construction of Self-dual Chern–Simons Vortices

In this note we discuss the solvability of Liouville-type systems in presence of singular sources, which arise from the study of non-abelian Chern Simons vortices in Gauge Field Theory and their asymptotic behaviour (for limiting values of the physical parameters). This investigation has contributed towards the understanding of singular PDE ’s in Mean Field form, in connection to surfaces with conical singularities, sharp Moser–Trudinger and log(HLS)-inequalities, bubbling phenomena and point-wise profile estimates in terms of Harnack type inequalities. We shall emphasise mostly the physical impact of the rigorous mathematical results established and mention several of the remaining open problems.     

Uniqueness of selfdual periodic Chern–Simons vortices of topological-type

In analogy with the abelian Maxwell–Higgs model (cf. Jaffe and Taubes in Vortices and monopoles, 1980) we prove that periodic topological-type selfdual vortex-solutions for the Chern–Simons model of Jackiw–Weinberg [Phys Rev Lett 64:2334–2337, 1990] and Hong et al. Phys Rev Lett 64:2230–2233, 1990 are uniquely determined by the location of their vortex points, when the Chern–Simons coupling parameter is sufficiently small. This result follows by a uniqueness and uniform invertibility property established for a related elliptic problem (see Theorem 3.6 and 3.7). Research supported by M.I.U.R. project: Variational Methods and Nonlinear Differential Equations.     

Uniqueness and symmetry results for solutions of a mean field equation on 𝕊2 via a new bubbling phenomenon

Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere 𝕊² involving a Dirac distribution supported at a point P ∈ 𝕊². Consistently with the physical applications, we show that solutions “concentrate” precisely around the point P for some limiting value of a given parameter. We use this fact to obtain symmetry (about the axis ) and uniqueness property for the solution. The presence of the Dirac measure makes such a task particularly delicate to handle from the analytical point of view. In fact, the bubbling phenomenon about the singularity allows the existence of solution sequences with a double‐peak profile near P. The new and more delicate part of this paper is to exclude this possibility by using the method of moving planes together with the Alexandrov‐Bol inequality. © 2011 Wiley Periodicals, Inc.     

Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

Motivated by the study of multivortices in the Electroweak Theory of Glashow–Salam–Weinberg [33], we obtain a concentration-compactness principle for the following class of mean field equations: on M, where (M,g) is a compact 2-manifold without boundary, 0 < a≤K(x)≤b, x∈M and λ > 0. We take with α _i > 0, δ _{p i} the Dirac measure with pole at point p _i ∈M, i= 1,…,m and ψ∈L ^∞(M) satisfying the necessary integrability condition for the solvability of (1)_λ. We provide an accurate analysis for solution sequences of (1)_λ, which admit a “blow up” point at a pole p _i of the Dirac measure, in the same spirit of the work of Brezis–Merle [11] and Li–Shafrir [35]. As a consequence, we are able to extend the work of Struwe–Tarantello [49] and Ding–Jost–Li–Wang [21] and derive necessary and sufficient conditions for the existence of periodic N-vortices in the Electroweak Theory. Our result is sharp for N= 1, 2, 3, 4 and was motivated by the work of Spruck–Yang [46], who established an analogous sharp result for N= 1, 2. Received: 24 September 2001 / Accepted: 7 December 2001     

A quantization property for blow up solutions of singular Liouville-type equations

Motivated by the study of self-dual vortices in gauge field theory (cf. (Vortices and Monopoles, Birkhauser, Boston, 1980; Solitons in Field Theory and Nonlinear Analysis, Monographs in Mathematics, Springer, New York, 2001)), we consider a “concentrating”solution-sequence u_k satisfying

     

Radial symmetry and symmetry breaking for some interpolation inequalities

We analyze the radial symmetry of extremals for a class of interpolation inequalities known as Caffarelli?CKohn?CNirenberg inequalities, and for a class of weighted logarithmic Hardy inequalities which appear as limiting cases of the first ones. In both classes we show that there exists a continuous surface that splits the set of admissible parameters into a region where extremals are symmetric and a region where symmetry breaking occurs. In previous results, the symmetry breaking region was identified by showing the linear instability of the radial extremals. Here we prove that symmetry can be broken even within the set of parameters where radial extremals correspond to local minima for the variational problem associated with the inequality. For interpolation inequalities, such a symmetry breaking phenomenon is entirely new.     

On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

In this paper we determine the interaction of diagonal defect clusters in regions of an Aztec rectangle that scale to arbitrary points on its symmetry axis (in earlier work we treated the case when this point was the center of the scaled Aztec rectangle). We use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary. In several of the treated situations this seems not to be accomplishable by previous methods. Our applications include the case of two long neutral strings, which turn out to interact by an analog of the Casimir force, two families of neutral doublets that turn out to interact completely independently of one another, a neutral doublet and a very long neutral string, a general collection of macroscopically separated monomer and separation defects, and the case of long strings consisting of consecutive monomers.