External Problems of Nonlinear Elliptic k-Hessian Equations

Completely nonlinear elliptic equations are often associated with problems with geometric backgrounds. The most typical work was to solve the Minkowski problem in convex bodies in the 1970s and to prove Qiu Chengtong's Calabi conjecture in Kahler geometry, both of which were reduced to the solvability of the Monge Ampere equation on a compact manifold. In the 1980s, Caffarelli Nernberg Spruck and Kohn published a series of five papers, starting to study completely nonlinear elliptic equations, especially the most important model, the k-Hessian (k=n is the Monge Ampere, k=1 is the Laplace operator) equation (boundary value problem).

We will report on the progress of subsequent elliptical k-Hessian. In the case of compact manifolds, such as existence problems on spherical or Kahler manifolds, Neumann boundary value problems on convex domains, or high-order Yamabe problems. Especially the existence and regularity of recent external problems.

Reported by: Ma Xinan, male, born in 1969, currently a professor at the School of Mathematical Sciences, University of Science and Technology of China. Ma Xinan graduated with a Ph.D. from the Mathematics Department of Hangzhou University in 1996. From 1996 to 2005, he worked in the Mathematics Department of East China Normal University. Since 2005, he has been teaching at the School of Mathematical Sciences, University of Science and Technology of China, mainly engaged in research on elliptic partial differential equations and geometric analysis.

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