Vortex condensation in the Chern-Simons Higgs model: An existence theorem

Caffarelli, Luis A.; Yang, Yi

doi:10.1007/bf02101552

Cited by 228 publications

(198 citation statements)

References 29 publications

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“…Sometimes called Liouville equation after [24], this problem and qualitatively similar ones have attracted great attention over the last decades. In a twodimensional domain or a compact manifold this type of equation arises in a broad range of applications, in particular in astrophysics and combustion theory, see [10,19,26] and references, the prescribed Gaussian curvature problem, [21,12,13], mean field limit of vortices in Euler flows, [8,14], and vortices in the relativistic Maxwell-Chern-Simons-Higgs theory [6,9,28,23].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Singular limits in Liouville-type equations

2005

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Abstract. We consider the boundary value problem ∆u + ε 2 k(x) e u = 0 in a bounded, smooth domain Ω in R 2 with homogeneous Dirichlet boundary conditions. Here ε > 0, k(x) is a non-negative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exactly m points as ε → 0 and satisfies ε 2 Ω ke u ε → 8mπ. In particular, we find that if k ∈ C 2 (Ω), infΩ k > 0 and Ω is not simply connected then such a solution exists for any given m ≥ 1

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Singular limits in Liouville-type equations

2005

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“…Equation (E u ) λ has been studied by Kazdan and Warner [29] in connection with the prescribed Gauss curvature problem, while, it also arises from some Chern-Simons Higgs model as discussed in Taubes [40,41], Hong, Kim and Pac [27], Jackiw and Weinberg [28], Spruck and Yang [38], Caffarelli and Yang [11], Tarantello [39], Struwe and Tarantello [35], Ding, Jost, Li and Wang [22,23], and the references therein. Related problems are studied by Carleson and Chang in [14].…”

Section: Introductionmentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

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Abstract:A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern-Simons Higgs model and have been studied recently by various authors. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

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“…We present some details about the PDE structure of the problem. In particular, we show that the existence proof in the compact setting case relies on our understanding of an interesting nonlinear elliptic equation arising in condensed-matter physics, namely, the self-dual Chern-Simons equation [29,34,37], which may be solved by minimizing an energy functional subject to an inequality constraint [21,87]. Section 5 contains some concluding comments.…”

Section: Prescribed Gauss Curvature Equationmentioning

confidence: 99%

Geometry, Topology, and Gravitation Synthesized by Cosmic Strings

Yang¹

2007

Pure and Applied Mathematics Quarterly

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In the context of static cosmic strings, the Einstein equations coupled to the Abelian Higgs model and the gauged σ-model reduce into a single equation, which directly relates the Gauss curvature of a Riemann surface hosting nontrivial geometry and gravitation to the energy density of the coupled matter-gauge sector. This equation gives rise to an equivalence relation between the topology of the Riemann surface and the topology of the complex line bundle which hosts and is determined by the physical interactions of the matter-gauge sector. The existence of solutions realizing a prescribed distribution of cosmic strings resembles the classical prescribed Gauss curvature problem.

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Vortex condensation in the Chern-Simons Higgs model: An existence theorem

Cited by 228 publications

References 29 publications

Singular limits in Liouville-type equations

Singular limits in Liouville-type equations

Harnack Type Inequality: the Method of Moving Planes

Geometry, Topology, and Gravitation Synthesized by Cosmic Strings

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