Gabriella Tarantello

Profile of Blow-up Solutions to Mean Field Equations with Singular Data

Abstract Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.

Vortices in the Maxwell-Chern-Simons Theory

Our aim is to prove rigorously that the Chern-Simons model of Hong, Kim, and Pac [13] and Jackiw and Weinberg [14] (the CS model) and the Abelian Higgs model of Ginzburg and Landau (the AH model, see [15]) are unified by the Maxwell-Chern-Simons theory introduced by Lee, Lee, and Min in [16] (MCS model). In [16] the authors give a formal argument that shows how to recover both the CS and AH models out of their theory by taking special limits for the values of the physical parameters involved. To make this argument rigorous, we consider the existence and multiplicity of periodic vortex solutions for the MCS model and analyze their asymptotic behavior as the physical parameters approach these limiting values. We show that, indeed, the given vortices approach (in a strong sense) vortices for the CS and AH models, respectively. For this purpose, we are led to analyze a system of two elliptic PDEs with exponential nonlinearities on a flat torus. c 2000 John Wiley & Sons, Inc.

Chern–Simons vortices in the Gudnason model

Abstract We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the N = 2 supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern–Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an inequality-constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.

On Singular Liouville Systems

We discuss a class of planar systems of Liouville type in presence of singular sources. When the coupling matrix admits positive entries, we provide necessary and sufficient conditions for the existence of radial solutions and corresponding uniqueness. For this purpose we point out a log HLS inequality in system’s form that involves weights and holds in the radial setting.

On Some Elliptic Problems in the Study of Selfdual Chern-Simons Vortices

In these lectures we use an approach introduced by Taubes (cf. [JT]) in the study of selfdual vortices for the abelian-Higgs model, in order to describe vortex configurations for the Chern-Simons (CS in short) theory discussed in [D1]. Notice that the abelian-Higgs model corresponds (in a non-relativistic context) to the bi-dimensional Ginzburg-Landau (GL in short) model (cf. [GL], 1[DGP]), for which much has been accomplished in recent years also away from the selfdual regime. In this respect, beside the seminal work of Bethuel- Brezis-Helein (cf. [BBH]), we mention for example: [BeR], [JS1],[JS2], [Lin1], [Lin2], [LR1], [LR2], [PiR], [PR] and the recent monograph by Sandier-Safarty [SS]. However, the methods and techniques introduced for the GL-model do not seem to apply as successfully for the CS-model (see the attemps of Kurzke-Sprin [KS1], [KS2] and Han-Kim in [HaK]). Thus, so far a rigorous mathematical analysis of CS-vortices has been possible only at the selfdual regime where Taubes approach applies equally well and allows one to reduce the vortex problem to the study of elliptic problems involving exponential nonlinearities. In this way it has been possible to treat many relevant selfd- ual theories of interest in theoretical physics by means of nonlinear analysis, see [Y1].

Non-topological Vortex Configurations in the ABJM Model

In this paper we study the existence of vortex-type solutions for a system of self-dual equations deduced from the mass-deformed Aharony–Bergman–Jafferis–Maldacena (ABJM) model. The governing equations, derived by Mohammed, Murugan, and Nastse under suitable ansatz involving fuzzy sphere matrices, have the new feature that they can support only non-topological vortex solutions. After transforming the self-dual equations into a nonlinear elliptic

Sharp estimates for solutions of mean field equations with collapsing singularity

{2\times 2}

Selfdual gauge field vortices :: an analytical approach

2×2 system we prove first an existence result by means of a perturbation argument based on a new and appropriate scaling for the solutions. Subsequently, we prove a more complete existence result by using a dynamical analysis together with a blow-up argument. In this way we establish that any positive energy level is attained by a 1-parameter family of vortex solutions, which also correspond to (constraint) energy minimizers. In other words, we register the exceptional fact in a BPS-setting that, neither a “quantization” effect nor an energy gap is induced upon the system by the rigid “critical” coupling of the self-dual regime.