Ryan Hynd (MIT/University of Pennsylvania), Analysis Seminar

Title: Extremal functions for Morrey’s inequality in convex domains

Abstract: A celebrated result in the theory of Sobolev spaces is Morrey’s inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor.  We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.