PDE/Analysis Seminar

Speaker: Josef Greilhuber Title: Polynomial bounds for the size of the critical set of harmonic functions Abstract: It is well-known that the complexity of a solution to an elliptic PDE is intricately related to the spatial growth rate of the solution. An instance of this phenomenon is the fact that one can bound the size of the zero set of a solution by a power of the solution's 'doubling index', a global quantity measuring growth. Obtaining analogous bounds for the size of the critical set turns out to be a much more delicate problem, and the best currently available bounds grow superexponentially in the doubling index. We will explore the reasons behind this difficulty and sketch a proof of a polynomial bound for the critical set of harmonic functions on a real-analytic manifold. This is joint work (in progress) with Benjamin Foster, Eugenia Malinnikova and Fedor Nazarov.