Mathematical Analysis and Applications of Concentration Phenomena in Nonlinear Elliptic Equations

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The primary goal of my proposal is to study nonlinear elliptic equations and systems with particular emphasis on problems with solutions that exhibit concentration phenomena. There are three main types of concentration phenomena depending on the nature of the nonlinearity: spikes, transition layers, and vortices. Spikes arise in Nonlinear Schrodinger equations (NLS), reaction diffusion sytem in mathematical biology, etc. Transition layers are associated with Allen-Cahn equation and phase trasitions in material sciences. Vortices play an important role in Superconductivity and Particle Physics modeled by the magnetic Ginzburg-Landau equation, Chern-Simons-Higgs system and Yang-Mills-Higgs system. There are two parts of this proposal. In the first part (pure mathematics part), we would like to understand the structure of entire solutions to semilinear elliptic PDEs modeling concentration phenomena. A major aspect of this part is to bring ideas from Differential Geometry into the analysis and construction of entire solutions for three important equations: the Allen-Cahn equation, the nonlinear Schrodinger equation and magnetic Ginzburg-Landau equation. These equations are typical representatives of semilinear elliptic problems. The objective is to establish an intricate correspondence between the study of entire solutions of some scalar equations and the theories of minimal surfaces, constant mean curvature (CMC) surfaces and Toda systems. In the second part (the applied mathematics part), we plan to investigate how different techniques and results in nonlinear PDEs, scientific computing, matched asymptotics, variational methods, critical point theory, dynamical systems, differential geometry and algebraic geometry can be applied to solve nonlinear equations arising from the physical world. Such problems include interface behavior in diblock copolymer theory, and localized pattern formation problems modeled by reaction-diffusion systems, with applications to biological morphogenesis, theoretical chemistry, hot-spot patterns of urban crime, etc.