The Monge-Ampere Operator in Analysis and PDEs

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本基金资助的研究活动包括蒙日-安培方程几何理论的以下几个方面：第一海森堡群中的正则理论；欧几里德空间中线性化Monge-Ampere算子解的正则性以及蒙日-安培方程解与凸势拟共形映射之间的联系。本文研究了拟共形和双lipschitz Jacobian问题的相关问题，以及一类拟共形映射的凸势的Monge-Ampere测度的刻画，以及相应的对亚椭圆情况的推广。蒙日-安培算子是无所不在的拉普拉斯算子的非线性近亲，拉普拉斯算子对物理和工程的巨大重要性早已得到充分证明。作为一种非线性算子，蒙日-安培算子要比拉普拉斯算子复杂得多。然而，当作用于凸函数时，它产生几何和测量理论对象，有助于澄清其工作原理。长期以来，蒙日-安培算符在微分几何和最佳质量传输中起着关键作用。最近，它被应用于设计反射天线和所谓的准共形映射。蒙日-安培方程的另一个令人兴奋的方面是它的线性化，它本身就可以应用于锋面形成和流体动力学的拉格朗日模型。线性化的蒙日-安培算子是一个具有挑战性的对象，因为它不属于众所周知的“一致椭圆”算子的范畴。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The research activities supported by this grant include the following aspects of the geometric theory for the Monge-Ampere equation: a regularity theory in the first Heisenberg group; regularity of solutions to the linearized Monge-Ampere operator in Euclidean space; and the emerging connections between solutions to the Monge-Ampere equation and quasiconformal mappings with convex potential. Problems related to the quasiconformal and bi-Lipschitz Jacobian problems, the characterization of certain quasiconformal mappings with convex potentials in terms of their Monge-Ampere measure, and the corresponding extensions to the subelliptic case will be investigated.The Monge-Ampere operator is the nonlinear cousin of the omnipresent Laplace operator, whose enormous importance for physics and engineering has long been well documented. As a nonlinear operator, the Monge-Ampere operator is far more complicated than the Laplacian. However, when acting on convex functions, it produces geometric and measure-theoretic objects that help to clarify its workings. For a long time, the Monge-Ampere operator has played a key role in differential geometry and optimal mass transportation. Somewhat more recently, it has found applications to the design of reflector antennas and to so-called quasiconformal mappings. Another exciting aspect of the Monge-Ampere equation comes from its linearization, which has applications in its own right to Lagrangian models of front formation and fluid dynamics. The linearized Monge-Ampere operator is a challenging object, for it does not fall into the category of the well-known "uniformly elliptic" operators.