Unique Continuation and Regularity of CR Mappings

主要部分的数学研究项目Shiferaw Berhanu涉及调查的有效性唯一延续的解决方案的系统的一阶偏微分方程和二阶偏微分方程。该项目包括问题的正则性某些映射之间的子流形在复杂的空间。研究中的二阶偏微分方程出现在电磁辐射、光学、地震学和声学的研究中。一些方程，要调查的是相关的固体力学，它们可以用来模拟弹性静态变形。它们也与流体力学有关，因为它们可以用来描述不可压缩粘性流体的运动。该项目的成果对多复变函数论、CR几何、线性和非线性偏微分方程有重要的应用。该项目将为研究生和年轻的研究人员提供几个有趣的问题。第一个主要问题是关于理解两个Cauchy-Riemann子流形上的几何条件，这些条件保证它们之间的CR映射在一点处消失到无限阶的唯一连续性。第二个问题是关于真实的二阶或高阶解析椭圆型偏微分方程解在边界上的唯一延拓。第三个问题涉及Cauchy-Riemann子流形之间CR映射的正则性。所采用的方法包括解析圆盘理论、非线性傅立叶变换（FBI变换）以及对研究中的二阶和高阶算子的绿色函数的精确分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main parts of the mathematics research project by Shiferaw Berhanu involve the investigation of the validity of unique continuation for solutions of systems of first order partial differential equations and second order partial differential equations. The project includes problems on the regularity of certain mappings between submanifolds in complex spaces. The second order partial differential equations under study arise in the study of electromagnetic radiation, optics, seismology, and acoustics. Some of the equations that are to be investigated are relevant to solid mechanics where they can be used to model elasto-static deformations. They are also of relevance in fluid mechanics since they can be employed to describe the motion of an incompressible viscous fluid. Results from the project have important applications to function theory of Several Complex Variables, CR Geometry, and linear as well as nonlinear partial differential equations. The project will provide several interesting problems to graduate students and young researchers.The first main problem concerns understanding geometric conditions on two Cauchy-Riemann submanifolds that guarantee unique continuation for a CR mapping between them that vanishes to infinite order at a point. The second problem concerns the unique continuation at the boundary for solutions of real analytic, second order or higher order, elliptic partial differential equations. The third problem involves the regularity of CR mappings between Cauchy-Riemann submanifolds. The methods to be employed include the theory of analytic discs, nonlinear Fourier transforms (FBI transforms), and a precise analysis of Green's functions for the second and higher order operators under study.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.