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Nonlinear Analysis

Non-simple Blowup solutions and vanishing estimates for singular Liouville equations

Lei Zhang - University of Florida, Gainesville

Location:  Hill Center Room 705
Date & time: Tuesday, 07 February 2023 at 1:40PM - 2:40PM

Abstract: The singular Liouville equation is a class of second order elliptic partial differential equations defined in two dimensional spaces: \[Delta u+ H(x)e^{u}=4pi gamma delta_0 \]where \(H\) is a positive smooth function, \(gamma>-1\) is a constant and \(delta_0\) stands for a singular source placed at the origin. It is well known that this equation is related to Nirenberg problem and is a reduction of Toda system which comes from a rich background. When a sequence of solutions tends to infinity near the origin, they are called blowup solutions. If \(gamma\) is a multiple of \(4pi\), the blowup solutions may violate spherical Harnack inequality around the origin, in which case they are called ?``non-simple blowup solutions". There were quite a few major challenges related to non-simple blowup solutions. In this talk I will report my recent joint works with Juncheng Wei and Teresa D'Aprile. Our results lead to new understanding not only on the structure of solutions of the singular Liouville equation, but also on Toda systems connected with Lie Algebra and Algebraic Geometry.

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