Singularity and regularity for Monge-Ampere type equations

Monge-Ampere 型方程的奇异性和正则性

• 批准号: DP230100499
• 负责人: A/Prof Jiakun Liu
• 金额: $ 28.41万
• 依托单位: University of Wollongong
• 依托单位国家: 澳大利亚
• 项目类别: Discovery Projects
• 财政年份: 2023
• 资助国家: 澳大利亚
• 起止时间: 2023-01-01 至 2025-12-31
• 项目状态: 未结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP230100499
• 关键词: Singularity; regularity; Monge; Ampere; type

The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.

蒙日-安培方程作为一个重要的非线性偏微分方程，在几何、物理和最优运输等多个领域都有应用。许多重要的问题和应用都与解的正则性有关，而解的正则性受到奇点的阻碍。本项目旨在对奇异集的几何进行分类，并采用创新的方法和发展偏微分方程的前沿技术，建立一般蒙日-安培型方程的综合正则性理论。预期的结果包括解决悬而未决的问题。该项目将显著提高澳大利亚在数学和应用这一重要领域的领导地位和专业知识。