Topics in Nonlinear Subelliptic Partial Differential Equations

本研究项目的重点是一类非线性偏微分方程（PDE），它们是平均曲率流PDE的亚黎曼类似物，最小表面PDE和保形n-拉普拉斯方程。与他的学生和合作者一起，首席研究员将建立在他之前与Mario Bonk和Giovanna Citti的工作基础上，以调查方程解的存在性，规律性和唯一性问题，并探索这些解的定性特征（例如，凸性，自相似性，渐近行为）。在与Bonk， Scott paul和Jeremy Tyson的合作中，首席研究员将研究亚黎曼空间中子流形的微分几何，从而为PDE理论提供几何背景。首席研究员还将继续与Michael Cowling和Loredana Lanzani合作，研究严格伪凸集之间生物全纯映射边界正则性的几何函数理论/非线性PDE方法。自“现实世界”的问题和应用开始以来，亚黎曼几何的研究一直受到推动。粗略地说，亚黎曼空间是那些结构可以被视为“受限几何”的空间：运动只能沿着给定的一组方向进行，这些方向在点与点之间变化，但仍然保证了全局可达性。亚黎曼几何的范围非常广泛，包括以下各种领域的应用：复杂分析，机器人手臂运动和控制理论，双曲几何，量子计算，卫星，量子力学，哺乳动物视觉皮层的结构功能。在这种情况下，偏微分方程的研究允许人们创建现实生活中的数学模型。然后使用这些模型来更好地理解所研究的系统，从而对其未来的行为做出准确的预测。

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.