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

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Geometric Analysis and PDEs

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1977))

Abstract

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].

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  1. Universita’ di Roma Tor Vergata- Dipartimento di Matematica, via della ricerca scientifica, 00133, Rome, Italy

    Gabriella Tarantello

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  1. Dept. Mathematics, Princeton University, Washington Rd., Princeton, 08544-1000, U.S.A.

    Sun-Yung Alice Chang

  2. Department of Mathematics, SISSA, Via Beirut 2-4, Trieste, 34014, Italy

    Antonio Ambrosetti

  3. Studi Avanzati (SISSA), Scuola Internazionale Superiore di, via Beirut 2-4, Trieste, 34014, Italy

    Andrea Malchiodi

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Tarantello, G. (2009). On Some Elliptic Problems in the Study of Selfdual Chern-Simons Vortices. In: Chang, SY., Ambrosetti, A., Malchiodi, A. (eds) Geometric Analysis and PDEs. Lecture Notes in Mathematics(), vol 1977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01674-5_4

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