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Title: Geometric Methods in Obstacle-Type Free Boundary Problems I

Abstract: Obstacle-type free boundary problems naturally appear as mathematical models in science and engineering with some particular motivations arising from contact problems in elasticity, options pricing in financial mathematics, and phenomenological models in superconductor physics. The first talk will focus on geometric methods that have been used to study regularity estimates in Obstacle-Type Free Boundary Problems. The regularity theory for obstacle-type problems (and other type of free boundary problems as well) was much inspired by the regularity theory for minimal surfaces. We will discuss the basic existence, uniqueness and regularity questions in the classical obstacle problem. We will point out generalizations and current problems of interest in this field of research. In the second talk we will focus on an obstacle-type problem arising in stochastic impulse control theory that appeared first as a model for cash management and portfolio optimization under transaction costs. Here the underlying theory for the obstacle problem has to be suitably modified to consider obstacle problems with an implicit and nonlocal obstacle. Regularity estimates will be presented and natural directions for future research discussed.burn