Grant: Discovery Grants Program - Individual

The goal of this proposal is to make significant progress in the understanding of the qualitative properties of Nonlinear Partial Differential Equations (PDEs) through a number of collaborations with world leaders on each topic. The theory of PDEs is one of the most fundamental branches of pure mathematics that connects a variety of fields including mathematical analysis and differential geometry. PDEs are present in almost all applications of mathematics, in which they provide a natural mathematical description of phenomena in the physical, natural and social sciences. Roughly speaking, the motion of every particle in nature can be modelled and analyzed by certain PDEs. Most of the PDEs that accurately model natural phenomena are, however, nonlinear. In general, there is no way of writing these solutions explicitly. Nonetheless, it is often possible to investigate the qualitative and quantitative properties of the unknown solutions of PDEs and models arising in Geometry and Mathematical Analysis, Mathematical Biology and Electrical/Mechanical Engineering. This proposal aims to respond to the need in this crucial interdisciplinary research area. From the Geometry and Mathematical Analysis perspective, this proposal is centred on the De Giorgi's conjecture (1978) that was inspired by the Bernstein's problem (1910) in the topic of differential geometry (minimal surfaces). I would like to request funding to work on a few generalizations of the De Giorgi's conjecture, as well as the conjecture itself in dimensions four to eight. Moreover, I am interested in developing the regularity theory for nonlinear eigenvalue problems for all powers (integer and non-integer) of the Laplacian operator. I plan to have collaborations with Nassif Ghoussoub and Juncheng Wei (UBC) who are experts in this field. From the Mathematical Biology perspective, we are interested in analyzing various biological models of PDEs. A significant part of our interest is to establish methods and theories for traveling waves, spreading speeds and propagation phenomena for impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. Mathematically, these models (PDEs) are very interesting because they include both differential and difference equations. In this topic, I plan to collaborate with my colleagues, Mark Lewis and Hao Wang, from The Centre for Mathematical Biology. From the Engineering perspective, we are interested in examining certain aspects of a few models (PDEs) arising in Mechanical and Electrical Engineering. In this direction of research, I would like to establish various Liouville theorems for stationary compressible Navier-Stokes equations. These equations describe the motion of fluid substances. I would like to initiate collaborations with Tai-Peng Tsai and Dong Li (from UBC), Dongho Chae (from Chung-Ang University) and Gregory Seregin (from Oxford). The other model that is of our interest is the Mircro-Electro-Mechanical System (MEMS) model. One of the primary goals in the design of MEMS devices is to optimize the pull-in distance over a certain allowable voltage range that is set by the power supply. Mathematically speaking, MEMS models are interesting because the nonlinearity appearing in these models (PDEs) is inverse square type that is singular. Regarding this model, I would like to launch a joint project with Mojgan Daneshman from the Department of Electrical and Computer Engineering (ECE) to carefully analyze the model.