Research Seminar in Mathematics - Uniqueness of extremals for Morrey's inequality
16 May 2018 13:00 T141, Teknikhuset
Speaker
Dr. Erik Lindgren from Stockholm University
Abstract
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes in particular that for a bounded domain and p>n, the sup-norm of a W_0^{1,p}-function can be bounded in terms of the L^p-norm of its gradient. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is the whole space or a ball).
I will discuss uniqueness properties of extremals of this inequality. These extremals are minimizers of a nonlinear Rayleigh quotient. In particular, I will present the result that in convex domains, extremals are determined up to a multiplicative factor. I will also explain why convexity is not necessary and why stareshapedness is not sufficient for this result to hold.