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Geometry-Topology Seminar

Thursday, February 7, 2019 - 4:30pm

Francis Seuffert



University of Pennsylvania


   The seminorm form of Morrey’s inequality states that the gradient L^p norm of a function on n-dimensional Euclidean space bounds its 1-n/p Holder seminorm (up to some fixed constant depending on p and n) when p>n. This inequality was (essentially) proven 80 year ago by C. B. Morrey Jr. However, until recently, nothing was known about extremals or the sharp constant of Morrey’s inequality. In a recent project, R. Hynd and I proved the existence of extremals and some of their qualitative characteristics. The key to our results is to show that a function, v, is an extremal of Morrey’s Inequality if and only if it satisfies a special PDE. In a recent project, we show that any given extremal, v, of Morrey's Inequality has a unique pair of points, x and y, where it achieves its 1−n/p Holder seminorm, they are the points where v achieves its absolute maximum and minimum, and v is analytic except at that pair of points. Moreover, using the PDE, we are able to show that extremals of Morrey’s inequality are cylindrically symmetric (if n > 2) or evenly symmetric (if n = 2) about the line containing x and y, reflectionally antisymmetric (up to addition by a constant), and unique up to operations that are invariant on the ratio of the seminorms in Morrey's Inequality. We also give explicit solutions for extremals when n = 1 and some numerical approximations of extremals for n = 2 and p = 4. This work is a collaboration with R. Hynd.