Symmetry results for minimizers of the Caffarelli-Kohn-Nirenberg inequalities and their relation with Onofri's inequality


In dimension N=2 many functional inequalities involving the Laplacian break down, like the Sobolev or the Hardy inequalities. Here we will study symmetry of minimizers for the so-called Caffarelli-Kohn-Nirenberg inequalities which still hold in dimension 2. We show that in some well chosen scaling, there is an unexpected relation between these inequalities and a particular extension of Onofri's inequality. This allows us to give a new proof for symmetry breaking for certain ranges of parameters.

On the other hand, we also prove symmetry of minimizers in cases in which no result was known so far, by a method not involving  moving or sliding planes arguments.

(This is work done in collaboration with Jean Dolbeault and Gabriella Tarantello).