Non-linear partial differential equations, free boundary problems and fractional operators

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This project deals with several aspects of the mathematical analysis of nonlinear partial differential equations. A large part of the endeavor is devoted to the study of elliptic and parabolic free boundary problems. The principal investigator is interested in the existence and regularity theory for such problems, and special attention will be paid to investigating the asymptotic behavior of the solutions (such as the long-time behavior and homogenization limits). Another direction of research concerns nonlocal elliptic and parabolic equations, mainly equations involving fractional Laplace operators. In particular, the project will investigate the effects of the nonlocality of these differential operators on well-known phenomena such as front propagation and homogenization. Finally, part of this project is devoted to certain nonlinear elliptic and parabolic equations of third and fourth order. The analysis of such equations is still poorly understood, mainly because of the lack of a maximum principle and other basic properties that their second-order counterparts possess. Developing existence and uniqueness theories for such equations will be one of the main challenges of the project.Nonlinear partial differential equations have applications in physics, biology, economics, engineering, and other areas of science. Free boundary problems, for instance, typically arise in the modeling of physical phenomena that involve interfaces (between two materials) that are changing with time (a good example is the study of the melting of a block of ice). Some free boundary problems discussed in this proposal turn up in the modeling of gas combustion and in the study of fluid flow. A better understanding of the effects of small inhomogeneities in the properties of the medium (for instance, impurities in the ice) is the goal of the "homogenization theory" mentioned earlier. Ultimately, this theory will allow the development of more accurate models to describe a great variety of phenomena. Nonlocal diffusion equations are another type of equation that are commonly used to model a broad range of phenomenon. The principal investigator proposes a new approach to deriving these equations using so-called microscopic (kinetic) models, for which the parameters can be easily related to simple physical quantities. This will lead to better accuracy in the models used by physical scientists. Finally, applications of higher order elliptic and parabolic equations are also numerous. The equations discussed in this proposal have some applications to the modeling of hydraulic fractures (which are used, for example, to propagate rock fractures in oil and gas reservoirs so as to enhance oil recovery) and to the study of biological membranes.