一类Monge-Ampère方程解的边界行为

项目名称： 一类Monge-Ampère方程解的边界行为

项目编号： No.11301231

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 吴亚东

作者单位： 江西师范大学

项目金额： 22万元

中文摘要： 仿射微分几何中, 完备双曲型仿射球对应一个在有界凸区域上的零边值方程, 相对微分几何中的一些曲率方程也归结到类似的蒙日-安培方程, 因此一个问题是如何刻画这些方程凸解的边界行为. 另一方面与这些方程解有联系的是凸区域上的特征函数, 它在区域边界附近有渐近展开式, 但展开式中系数的几何意义并不完全清楚. 本项目结合仿射几何与蒙日-安培方程的理论技巧, 研究用边界曲面的仿射不变量来表示特征函数展开式中的系数; 并对这类方程的凸解建立导数估计, 用来刻画解的边界行为及对应超曲面的整体性质.

中文关键词： Monge-Ampère 方程；边界行为；导数估计；特征函数；渐近展开

英文摘要： In affine differential geometry, a complete hyperbolic affine hypersphere corresponds to a zero-boundary value equation on a bounded convex domain , and in relative differential geometry, some curvature equations also correspond to similar Monge-Ampère equations. A problem is how to describe the boundary behaviors of their convex solutions. On the other hand, the characteristic functions on bounded convex domains have relations with these convex solutions, and have asymptotic expansions near the boundary, but the geometrical meaning of coefficients in the expansions are not good known. In this project we use the theory and methods of affine geometry and Monge-Ampère equation to study: using the affine invariants to describe the coefficients in the expansions of characteristic functions; giving derivative estimates of convex solutions of these Monge-Ampère equations. These results can be use to describe the boundary behaviors of solutions and the global properties of corresponding hypersurfaces.

英文关键词： Monge-Ampère equation；Boundary behavior；Derivative estimate；Characteristic function；Asymptotic expansion