Abstract
We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager’s description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.
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Communicated by P. Constantin
Research partially supported by FIRB project “ Analysis and Beyond” and by MIUR project “Metodi variazionali e PDE non lineari”.
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Bartolucci, D., Malchiodi, A. An Improved Geometric Inequality via Vanishing Moments, with Applications to Singular Liouville Equations. Commun. Math. Phys. 322, 415–452 (2013). https://doi.org/10.1007/s00220-013-1731-0
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DOI: https://doi.org/10.1007/s00220-013-1731-0