Unique Continuation and Regularity of Mappings and Functions in Several Complex Variables

Partial differential equations describe dynamic interrelationships between several mutually dependent quantities. As such, they are ubiquitous in the mathematical modelling of a host of systems in physics, engineering, and other scientific fields. This project will explore two salient issues related to the theory of partial differential equations, in the setting of higher-dimensional spaces defined via complex numbers. The first topic is the question of unique continuation, which – in rough terms – asks when infinitesimal data about the solution to a differential equation suffices to determine the behavior of the solution on macroscopic scale. The second topic is the regularity of solutions. In many physically relevant situations, partial differential equations exhibit a self-improving regularity property: solutions that are a priori assumed to exist within a broad class of functions with a weakly defined notion of smoothness, in fact can be shown a posteriori to satisfy a much more stringent smoothness condition. Such results are important both for the theory of partial differential equations and for scientific applications. The types of second order partial differential equations under consideration in this project are relevant for other equations of physical significance, such as the wave and heat equations, the Klein-Gordon equation, Maxell’s equation, and Schrödinger’s equation. The project will also contribute to the development of a scientifically literate workforce through graduate recruitment and training. The primary focus of the project is on the theory of CR mappings between CR manifolds. CR mappings are a foundational concept in the theory of several complex variables. In this context, a general unique continuation problem asks when a CR mapping between two embedded CR manifolds, that vanishes to infinite order at a single point, must vanish identically in a neighborhood of that point. Very few sufficient conditions are known, even in simple special cases. This question will be investigated in concert with related unique continuation problems for solutions of second order subelliptic partial differential equations with real analytic coefficients. The second part of the project involves the regularity theory of CR mappings on embeddable and abstract CR manifolds, along with the regularity of CR functions on abstract CR manifolds. Such results have applications to the theory of solutions of systems of first order complex nonlinear partial differential equations. The current project will focus on understanding conditions on the underlying manifolds that guarantee smoothness of CR mappings. 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.