Georgia Institute of Technology College of Sciences

This talk is centered around the symmetry properties of optimizers for the Caffarelli-Kohn-Nirenberg (CKN) inequalities, a two parameter family of inequalities. After a general overview I will explain some of the ideas on how to obtain the optimal symmetry region in the parameter space and will present an application to non-linear functionals of Aharonov-Bohm type, i.e., to problems that include a  magnetic flux concentrated at one point. These functionals are rotationally invariant and, as I will discuss, depending on the magnitude of the flux, the optimizers are radially symmetric or not.