Free Boundary Problems and nonlinear PDEs

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The principal investigator's past research has focused on interface problems arising in a variety of physical phenomena such as phase separation, fluid dynamics, materials science, and continuum limits of nonequilibrium particle systems. The goal of the current project is to gain a better understanding of general properties of free boundary problems using various notions of weak solutions, with the help of maximum-principle-type arguments, harmonic analysis, measure theory, and interacting particle systems. Specific questions under investigation are (i) global-time existence and uniqueness; (ii) uniting notions of weak solutions; (iii) regularity properties; (iv) long-time behavior of solutions and (v) geometric properties of free boundaries. Another project is to study homogenization of the free boundaries in periodic and random media, moving with oscillating normal velocity caused by inhomogeneities in the media. An example is the dynamics of water droplets spreading on an irregular surface. The lower dimensional nature of free boundaries bring major challenges to the homogenization, especially in nonvariational settings. The goal is to prove existence, uniqueness, and stability of the effective free boundary problem in periodic and random media.Nonlinear interface motions arise naturally in physical applications, and they have been extensively studied in the physics and applied mathematics literature. A classical example is the melting of ice pieces or snow crystals, where the central issue of interest is the evolution of the ice/water/air interface. The highly nonlinear structure and the development, in general, of finite-time singularities on the interface -- such as a stream of water pinching into droplets -- give rise to rather challenging difficulties in the rigorous analysis. It is important, both in theory and in practice, to understand what type of singularities occur and under what circumstances the behavior of the interfaces exhibits stability.