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

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Abstract

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.

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  1. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Gabriella Tarantello

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Correspondence to Gabriella Tarantello.

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This work is supported by PRIN12: “Variational and Perturbative Aspects in Nonlinear Differential Problem” and by FIRB project: “Analysis and Beyond”.

Lecture given at the Seminario Matematico e Fisico di Milano by Gabriella Tarantello on February 26, 2015, on the occasion of the Luigi and Wanda Amerio Prize 2014 tributed to her by the Istituto Lombardo Accademia di Scienze e Lettere.

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Tarantello, G. Analytical Issues in the Construction of Self-dual Chern–Simons Vortices. Milan J. Math. 84, 269–298 (2016). https://doi.org/10.1007/s00032-016-0259-0

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