Potential Theory of Functions of Bounded Variation and Quasiconformal Maps

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When we view images of objects, what we see are various locations of the object emitting different intensities of light. We can think of the image of the object as formed by a collection of surfaces of different shapes, each with its own uniform brightness. Similarly, in the study of various mathematical objects such as functions that measure temperature at different locations, functions that measure electro-magnetic intensities, and functions that measure the velocity of fluid particles at various places in a fluid, can be understood in terms of the shape of the level sets of the function. (A level set is a set where the function takes on a given constant value.) This project in metric space analysis explores the behavior of functions that arise in the study of potential theory and quasiconformal mappings in terms of level sets. Applications of the the research project include image processing and edge detection. Many components of this project are suitable dissertation material for graduate students, and therefore will contribute to the training of future members of the STEM workforce.This project is concerned with links between geometry of a metric space given in terms of its sets of finite perimeter on the one hand, and nonlinear potential theory and quasiconformal mappings on the other hand. The spaces considered are equipped with a doubling measure supporting a 1-Poincare inequality. In the first part of the project the PI will explore interactions between collections of sets of finite perimeter and quasiconformal mappings, and between collections of sets of finite perimeter and nonlinear potential theory. The second part of the project will explore "tangent space" regularity of sets of quasiminimal boundary surfaces. The third part of the project is to develop a potential theory for functions of bounded variation. The last part of the project is to obtain a characterization of certain Poincare inequalities in terms of modulus of families of sets of finite perimeter. Applications of the research include further understanding of connections between metric geometry related to sets of finite perimeter and solutions to certain nonlinear partial differential equations. Such sets arise in the study of image processing and edge detection, while abstract metric spaces arise in Riemannian manifolds theory when considering Gromov-Hausdorff limit spaces as found in the works of Cheeger, Gromov, and Perelman. Furthermore, these projects will expand the current knowledge about geometry and the theory of sets of finite perimeter in Carnot-Caratheodory spaces.