Regularity Theory for Elliptic Equations and Free Boundaries

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The aim of this project is to study the mathematical structures of various differential equations, concentrating on models that may develop moving interfaces. These problems are used to model a number of different phenomena in Physics, Mathematical Finance, Image processing, and Biology. The main mathematical challenge in such problems is to understand the smoothness or regularity of the interfaces. The models combine the difficulty of analyzing the differential equation with the difficulty in locating the moving interface.The nonlocal equations to be investigated in this research project arise naturally in probability, in the study of stochastic processes with jumps. More precisely, free boundary problems involving nonlocal equations appear when considering optimal stopping problems, which are used in the pricing of American options in finance. When the underlying stochastic process is Brownian motion (with continuous paths), the regularity theory for solutions and free boundaries is well understood. However, when considering jump-processes the equations become nonlocal, and the structure of the model is more complicated. One of the main goals of the project is to provide a full understanding of such models, establishing regularity results for solutions and free boundaries in this type of problems.