抛物型Monge-Ampere方程的外问题与多值解

项目名称： 抛物型Monge-Ampere方程的外问题与多值解

项目编号： No.11201343

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 代丽美

作者单位： 潍坊学院

项目金额： 22万元

中文摘要： Monge-Ampere方程有着很强的几何背景和重要的实际意义，对该方程的探讨一直是偏微分方程领域的热点。本项目主要研究一类抛物型Monge-Ampere方程的外问题和多值解。对于抛物型Monge-Ampere方程的外问题，本项目将首先构造外问题的粘性下解，然后利用Perron方法证明粘性解的存在性，进一步讨论解的正则性。对于抛物型Monge-Ampere方程的多值解，本项目首先给出抛物型方程多值解区域的几何背景，然后利用Perron方法证明多值解的存在性，结合抛物型Monge-Ampere方程本身的特点，解决多值粘性解的整体连续性。进一步利用Monge-Ampere方程的正则性理论讨论当"割口曲线"为"平面曲线"时解的光滑性。该研究将为其它类型的抛物型Monge-Ampere方程奠定基础。

中文关键词： 抛物型Monge-Ampere方程；外问题；多值解；对称性；数值解

英文摘要： The Monge-Ampere equations have very important situation in geometric and are significant in application. The discussion of Monge-Ampere equations has been a hot topic in partial differential equations. In this program, the exterior problem and the multi-valued solutions to parabolic Monge-Ampere equations will be mainly investigated. As for the exterior problem of parabolic Monge-Ampere equations, we will first construct the viscosity subsolution of the exterior problem and then obtain the existence of viscosity solutions using the Perron method. Furthermore, the regularity of viscosity solutions will be discussed. As for the multi-valued solutions to parabolic Monge-Ampere equations, we will first give the geometric situation of the multi-valued solutions to parabolic partial differential equations. Then we will get the existence of the multi-valued viscosity solutions to parabolic Monge-Ampere equations using the Perron method. Combined with the characteristic of parabolic Monge-Ampere equations themselves, the overall continuity of the multi-valued viscosity solutions will be solved. Furthermore, the smoothness of the multi-valued viscosity solutions will be discussed using the regularity theory of Monge-Ampere equations when the "cut curve" is "plane curve". This program will lay a foundation for the ot

英文关键词： parabolic Monge-Ampere equations；exterior problem；multi-valued solutions；symmetry；numerical solutions