The phase diagram of the Caffarelli-Kohn-Nirenberg inequalities

School of Mathematics Colloquium
Monday, February 22, 2016 - 4:00pm for 1 hour (actually 50 minutes)
Skiles 005
Michael Loss – School of Mathematics, Georgia Tech
Vladimir Koltchinskii
The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are {\it not} radial. In this talk I explain this and related problems andindicate a proof that, in the remaining parameter region, the optimizers are in fact radial. The novelty is the use of a flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.