This project involves the study of semilinear partial differential equations in the plane and systems of linear and nonlinear partial differential equations in higher dimensions. The semilinear equations are motivated by Vekua-type equations for the Cauchy-Riemann operator while the systems of equations are generalizations of the tangential Cauchy-Riemann equations on embedded CR submanifolds. The problems to be investigated include the analyticity of solutions of nonlinear partial differential equations and the properties of the solutions of Vequa-type equations when the Cauchy-Riemann operator is replaced by more general complex vector fields. The tools that may be used for the semilinear equations come from the solvability theory for complex vector fields in various function spaces and the theory of ordinary differential equations with periodic coefficients. For the nonlinear equations, the tools will include those developed in the theory of holomorphic extendability of CR functions including the FBI transform, analytic discs, and the Baouendi-Treves approximation theorem for vector fields with rough coefficients.
The research in this project is expected to have applications to partial differential equations and geometry. The semilinear equations arise from a geometrical problem that concerns the existence of nontrivial infinitesimal bendings for a given surface. This problem has physical applications to the elasticity of thin shells. The nonlinear equations arise in numerous geometrical and physical applications including in the modeling of atmospheric phenomena, and in the study of limit shapes of surfaces that minimize surface tension. The research activity on this project will be a source of meaningful problems for graduate students and recent Ph.D's.
The research activity from the project produced new regularity results for solutions of first order nonlinear complex partial differential equations. The regularity results yield as a consequence instability results with respect to a nonanalytic perturbation of an analytic initial datum in Cauchy problems for first order nonlinear partial differential equations. The project has also resulted in the development of a new class of nonlinear Fourier transforms that characterize the smoothness and analyticity of functions. These transforms can be applied to analyze the regularity of solutions of systems of first order partial differential equations with complex-valued coefficients. Another problem considered in the project concerns the boundary properties for the solutions of the semi linear Vekua's equation in the plane. The results that were obtained show that such solutions share many important boundary properties with holomorphic functions. It is also shown that a key regularity property that boundary values of holomorphic functions have is not shared by the boundary values of Vekua's equation. The project also led to results on a generalization of the classical Schwarz Reflection Principle of one complex variable to solutions of more general complex vector fields. A CR version of Bochner's extension theorem for tube CR manifolds where the base of the tube is a rather general set has also been established in the project. The results from the project have applications to function theory of several complex variables, geometry, and the theory of single and systems of first order nonlinear partial differential equations with complex coefficients. The insights gained in the research activity have been used in an essential way to recently solve two important conjectures on the smoothness of CR mappings between certain classes of CR manifolds. The semi linear equations that were studied in the project arise quite naturally from a geometrical problem that involves bending of surfaces and this problem in turn is closely connected to the stability of thin shells in elasticity. First order complex nonlinear partial differential equations are relevant to physical problems such as the modeling of atmospheric phenomena. They are also relevant to geometrical problems such as the study of limit shapes of surfaces that minimize surface tension. The research activity has generated meaningful and interesting problems for students and young researchers.
