Topics in Nonlinear Subelliptic Partial Differential Equations

非线性亚椭圆偏微分方程主题

• 批准号: 0800522
• 负责人: Luca Capogna
• 金额: $ 13.5万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800522&HistoricalAwards=false
• 关键词: Topics; Nonlinear; Subelliptic; Partial; Differential

The focus of this research project is a family of nonlinear partial differential equations (PDE) that are the sub-Riemannian analogues of the mean curvature flow PDE, the minimal surface PDE, and the conformal n-Laplacian. Together with his students and his collaborators, the principal investigator will build on his previous work with Mario Bonk and Giovanna Citti in order to investigate questions of existence, regularity, and uniqueness of solutions to the equations and to explore the qualitative features of these solutions (e.g., convexity, self-similarity, asymptotic behavior). In collaboration with Bonk, Scott Pauls, and Jeremy Tyson, the principal investigator will study the differential geometry of submanifolds in a sub-Riemannian space, thereby providing the geometric background for the PDE theory. The principal investigator will also continue his work with Michael Cowling and Loredana Lanzani and study a geometric function theory/nonlinear PDE approach to the boundary regularity of biholomorphic mappings between strictly pseudoconvex sets. The study of sub-Riemannian geometry has been driven since its origins by "real world" problems and applications. Roughly speaking, sub-Riemannian spaces are those whose structure can be viewed as a "constrained geometry": motion is possible only along a given set of directions, which changes from point to point yet nevertheless guarantees global accessibility. The scope of sub-Riemannian geometry is startlingly broad, including applications to areas as diverse and varied as the following: complex analysis, motion of robot arms and control theory, hyperbolic geometry, quantum computing, satellites, quantum mechanics, the structural function of the mammalian visual cortex. The study of partial differential equations in this setting allows one to create mathematical models of real-life situations. One then uses the models to achieve a better understanding of the systems under study and, as a result, to make accurate forecasts of their future behavior.

本研究的重点是一族非线性偏微分方程族，它们是平均曲率流偏微分方程组、极小曲面偏微分方程组和共形n-拉普拉斯偏微分方程组的次黎曼类似。与他的学生和他的合作者一起，首席研究员将在他之前与Mario Bonk和Giovanna Citti的工作的基础上，研究方程解的存在、正则性和唯一性问题，并探索这些解的定性特征(例如，凸性、自相似、渐近行为)。与Bonk，Scott Pauls和Jeremy Tyson合作，主要研究者将研究子流形在次黎曼空间中的微分几何，从而为PDE理论提供几何背景。主要研究人员还将继续他与Michael Cowling和Loredana Lanzani的工作，并研究严格伪凸集之间双全纯映射的边界正则性的几何函数论/非线性PDE方法。次黎曼几何的研究自诞生以来一直受到“真实世界”问题和应用的推动。粗略地说，次黎曼空间是那些其结构可以被视为“受限几何”的空间：运动只能沿着一组给定的方向进行，这组方向会逐点变化，但仍然保证了全局的可访问性。亚黎曼几何的范围令人惊讶地广泛，包括应用于各种不同的领域：复杂分析、机器人手臂的运动和控制理论、双曲几何、量子计算、卫星、量子力学、哺乳动物视觉皮质的结构功能。在这一背景下研究偏微分方程式使人们能够创建真实情况的数学模型。然后，人们使用这些模型来更好地理解所研究的系统，并因此对它们未来的行为做出准确的预测。