Regularity Questions in Linear and Nonlinear Partial Differential Equations

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The project focuses on mathematical research in certain partial differential equations (PDE) that model phenomena in physics, engineering, materials science, and economics. The study of PDE arising in models for composite materials, in particular fiber reinforced materials, is of increasing importance due to industry's need to design for improved performance. The mathematical research of the Monge-Ampére equation and related equations has particularly significant applications in differential geometry and optimal mass transport such as, for example, constructing surfaces with prescribed Gaussian curvature and reflector/refractor design. The principal investigator (PI) will carry out research closely related to these topics and will attempt to address some of the open questions in these areas. He will engage graduate students and postdoctoral researchers in the work of the project.The PI will focus his attention on several questions in three main topical areas. First, he will develop new methods to study elliptic and parabolic equations with mixed boundary conditions and rough coefficients in nonsmooth domains by using tools from harmonic analysis and conformal maps, parabolic equations with nonlocal time derivatives or more generally with nonlocal derivatives in both space and time, and Kolmogorov equations of ultraparabolic (or hypoelliptic) type with measurable coefficients. Second, the PI will study the regularity theory for degenerate fully nonlinear equations, in particular the m-Hessian equation with optimal power, the degenerate quotient Hessian equations, and more general types of Hessian equations with elementary symmetric polynomials. Finally, regarding PDE arising in the study of composite materials, the PI is particularly interested in composites with Lipschitz inclusions, and quasilinear or singular/degenerate equations of this type, and will develop new methods for the analysis of these PDE.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.