The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula

AbstractFor a bounded domain Ω⊂R2, we establish a concentration-compactness result for the following class of “singular” Liouville equations:−Δu=eu−4π∑j=1mαjδpj in Ω where pj∈Ω, αj>0 and δpj denotes the Dirac measure with pole at point pj, j=1,…,m. Our result extends Brezis–Merle's theorem (Comm. Partial Differential Equations16 (1991) 1223–1253) concerning solution sequences for the “regular” Liouville equation, where the Dirac measures are replaced by Lp(Ω)-data p>1. In some particular case, we also derive a mass-quantization principle in the same spirit of Li–Shafrir (Indiana Univ. Math. J.43 (1994) 1255–1270). Our analysis was motivated by the study of the Bogomol'nyi equations arising in several self-dual gauge field theories of interest in theoretical physics, such as the Chern–Simons theory (“Self-dual Chern–Simons Theories” Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin, 1995) and the Electroweak theory (“Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions,” World Scientific, Singapore)

OPTICAL AND FUNCTIONAL PERFORMANCE OF PHAKIC ANGLE-SUPPORTED INTRAOCULAR LENS FOR THE CORRECTION OF HIGH MYOPIA IN 18-MONTHS FOLLOW-UP STUDY

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An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

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.Comment: some references adde

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.Comment: to appear in GAF