This project concerns the analysis of nonlinear problems in elliptic partial differential equations, integro-differential equations and some related free boundary problems. We will focus on the obstacle problem for the fractional Laplacian, the thin obstacle problem, nonlinear integro-differential operators, and optimal design of composites. The obstacle problem for the fractional Laplacian is closely linked to lower dimensional obstacle problems, as the PI has recently shown in collaborative work. Thus these two problems will be investigated together. The PI also intends to generalize the concepts of viscosity solutions, well-posedness, and regularity results from the theory of fully nonlinear elliptic equations to nonlinear integro-differential equations. The PI intends to apply ideas from free boundary problems to some questions about composite design in materials science. The microstructures that maximize a certain effective quantity, or a combination of such, may sometimes be found by solving a free boundary problem. Analysis of this problem will provide a better understanding of these optimal structures.
The free boundary problems studied in this project arise in elasticity, financial mathematics, and composite material design. The obstacle problem is originally a model for the shape of an elastic membrane resting on a solid body. The same equation is used in financial mathematics for pricing American options. When stock prices have large jumps they may be modeled using discontinuous processes; this results in an obstacle problem for an integro-differential operator. The research proposed here studies the qualitative properties of the solutions of this problem as well as general nonlinear problems arising from stochastic games driven by random processes with jumps. Related ideas are also applied to the design of composite materials. Suitable free boundary problems enable the analysis of the microscopic structures that are optimal for certain purposes, such as to maximize the sum of the thermal and electric conductivity in a material.
