CAREER: Front Propagations and Viscosity Solutions

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This project concerns some nonlinear partial differential equations (PDEs) that appear naturally in physics, economics, and engineering and that arise, for example, in the study of crystal growth, composite materials, combustion, game theory, and optimal control theory. The equations studied have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, and inverse problems. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the PDEs under investigation. A key object of the research is a crystal growth model, in which the crystal grows in both horizontal direction by adatoms, and in vertical direction by nucleation in a supersaturated media. To make practical use of the model, it is extremely important to understand deeply the qualitative and quantitative aspects of the growth speed of the crystal. An integral part of the project is the educational component including bringing up the number of graduate PDE students at University of Wisconsin-Madison through various activities. The incoming graduate students interested in PDE are encouraged to participate in the principal investigator's PDE reading seminar, and to interact more with their peers and postdocs in the area. In term of undergraduate training, the principal investigator plans to increase the interest of University of Wisconsin-Madison undergraduate mathematics majors in the study of Analysis and PDE through some individual mentoring plans, and two undergraduate summer schools. Besides, the principal investigator plans to organize two conferences in nonlinear PDE and related topics for early career researchers.The proposed research involves two themes. The first is about a level-set mean curvature equation with driving and source terms, and applications to a crystal growth model, in which each level set of the unknown evolves in time by its mean curvature with unit constant force. The second involves Eikonal equations, and homogenization, where the zero-level set of the unknown moves in a periodic environment with highly oscillatory normal velocity. The principal investigator and his collaborators have recently developed some new approaches, which provided solutions to several open problems in these two themes and related areas. The new approaches are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of nonlinear PDE and viscosity solutions.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.