题目: Sobolev inequalities and regularites of linearized complex Monge-Ampere and Hessian equations
报告人: 周 斌 教授 （北京大学）
摘要: In this talk, we study the regularity of solution to the linearized complex Monge-Amp\`ere and Hessian equations when the complex $k$-Hessian is bounded from above and below. We first establish some estimates of Green's functions associated to the linearized equations. Then we prove a class of new Sobolev inequalities. With these inequalities, we use Moser's iteration to investigate the a priori estimates of Hessian equations and their linearized equations, as well as the K\"ahler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Amp\`ere and Hessian equations under an extra integrability condition on the coefficients. The approach works in both real and complex case.
报告人简介: 北京大学数学学院研究员，博士生导师，主要从事复几何，几何分析和完全非线性方程的研究。2012年获得澳大利亚基金会 Discovery Early Career Research Award奖，2018年获得国家级人才科学基金项目。已经在相关方向发表20多篇高质量的学术论文。