Academic Report
Title: Second Derivative Interior Estimation for a Class of Completely Nonlinear Equations
Reporter: QIU Guohuan (Chinese University of Hong Kong)
Time: May 2, 2019 (Thursday) AM 8:30-9:30
Location: A1101# room, Innovation Park Building
Contact: Prof. DAI Guowei (tel:84708351-8115)
Abstract: We prove Heinz's second derivative interior estimate for two-dimensional Mone-Ampere equation. But this result has Pogorelov's counterexample for the higher dimensional Mong-Ampere equation. Most completely non-linear equations do not have second derivative internal estimates because of Pogorelov's counterexamples. Warren-Yuan, however, proved the second-order interior estimate for a special quadratic Hessian equation by integral method. I find that this integration method is applicable to the general three-dimensional quadratic Hessian equation and the pre-determined curvature equation of hypersurfaces. The key is to find the corresponding monotonicity formula to transform point estimation into integral estimation. Then it is relatively easy to complete the integral estimation by using Hessian operator divergence structure.
The brief introduction to the reporter: Qiu Guohuan, Assistant Professor, Chinese University of Hong Kong, Bachelor of Mathematics, Northwest University, 2005.9-2009.6, Mashinan, Master Tutor of Chinese University of Science and Technology from September 2010 to June 2016, McGill University from August 2015 to August 2016, Research Assistant, Peking University of Science and Technology from June 2014 to June 2015, Post-doctoral, 2016.8-2018.8.8 McGill University, and Assistant Professor of Chinese University of Hong Kong from September 2018 to present. Several SCI papers have been published in Duke Math. J., Comm. Math. Phys., Int. Math. Res. Not. IMRN., J. Math. Study, SCHENTIA SINICA Mathematica, Commun.Contemp. Math.and other top international journals.
