Semilinear and nonlinear pdes motivated by complex variables and CR manifolds and the Bochner extension phenomenon

点击翻译按钮获取中文摘要

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.