黃信元 Hsin-Yuan Huang

Nonlinear PDEs in Geometry and Mathematical Physics

1. With Professor Zhang, we established the sharp estimates of multi-bubbles solutions on compact Riemann surface. In this work, we identified a robust property of these bubbling solutions, making their calculation highly intricate. Despite numerous unsuccessful attempts, in this paper, we successfully overcome all obstacles and completely solve the problem within the most general context.

2. I investigate the bubbling solutions of Liouville systems which show the concentration phenomena of solutions. In my work, I use a contraction mapping argument to prove the concentration phenomena. My result suggests that it is possible to use other arguments to obtain the sharp estimate results in the work of Lin-Zhang(2010 CPAM) without the assumption on the sign of the difference of parameter and limiting parameter.

3. In the collaboration with Professor Lin, we calculated the Leray-Schauder degree counting formulas of the corresponding curvature equation and obtained the form of the degree formula for k1 non-integer angles and k2 integer angles. Although the degree formula was obtained by Chen-Lin (CPAM2002, CPAM 2003), the explicit forms are not well-studied until this work. As an application of the degree counting formula, we can obtain the property of the monodromy matrices of the second order Fuchsian ODEs.