报告人简介
邱国寰博士，现任中国科学院数学与系统科学研究院研究员，2019年获得中国数学会钟家庆奖。2016年博士毕业于中国科学技术大学。曾在加拿大麦吉尔大学和香港中文大学从事研究工作。主要研究方向为偏微分方程和几何分析。相关研究论文发表在Duke Math J.,Amer.J. Math.,Comm.Math.Phys.等国际一流数学期刊上。
内容简介
We establish a priori interior curvature estimates for the special Lagrangian curvature equations in both the critical phase and convex cases. In dimension two, we observe that this curvature equation is equivalent to the equation arising in the optimal transportation problem with a "relative heat cost" function, as discussed in Brenier's paper. When 0 < Θ < π/2 (supercritical phase), the equation violates the Ma-Trudinger-Wang condition. So there may be a singular C^{1,a} solution in supercritical case which is different from the special Lagrangian equation. We have also demonstrated that these gradient estimates of these curvature equations hold for all constant phases. It is worth noting that for the special Lagrangian equation, particularly in subcritical phases, the interior gradient estimate remains an open problem. This is joint work with Xingchen Zhou.
