Geometric Properties of Second Order Elliptic Partial Differential Equations

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Many physical processes, for example the way heat spreads from a lit candle or a radiator throughout the room or the way two waves in a pond interact with each other, are rather well understood and we have several equations that describe the processes. There are usually three main questions that are being asked: (1) Is the equation correctly describing nature? (2) Does the equation actually have a solution? (3) Can the solution be computed, either by hand or on a computer? The purpose of this project is to investigate a question that is not asked quite as often: (4) what does the solution actually look like? If we are heating a room with, say, four candles and a radiator, which spot is going to be the coldest? These simple questions lead to both interesting and beautiful mathematics as well as surprising applications in practice (the Google Search Algorithm is essentially based on these types of structures, "cold" webpages are lower ranked than "hot" webpages).This project is dedicated to the study of (uniformly) elliptic second order partial differential equations, the main focus being on geometric properties of the solution and how those interact with the geometry of the underlying domain. Three explicit problems that will be studied are the (1) the location of extrema and critical points, (2) the geometry of level sets and (3) the geometry of eigenfunctions of an elliptic operator. The main types of applications will be (4) the analysis of spectral methods on graphs and (5) localization phenomena in mathematical physics. The main tools will be basic facts from geometry analysis to reinterpret analytic estimates geometrically and vice versa, the interpretation of elliptic equations as fixed points in time of an associated parabolic equation (as well as associated techniques from parabolic equations) and various aspects of spectral theory. Tools from the discrete world may be useful when reducing estimates to toy models in the graph setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.