The phase diagram of the Caffarelli-Kohn-Nirenberg inequalities

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 not radial. In this talk I indicate a proof that, in the remaining parameter region, the optimizers are in fact radial.

The proof will proceed via a well chosen flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.