Unique Continuation and Regularity of CR Mappings

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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.