  We analyze the radial symmetry of extremals for a class of interpolation
inequalities known as Caffarelli-Kohn-Nirenberg inequalities, and for a class
of weighted logarithmic Hardy inequalities which appear as limiting cases of
the first ones. In both classes we show that there exists a continuous surface
that splits the set of admissible parameters into a region where extremals are
symmetric and a region where symmetry breaking occurs. In previous results, the
symmetry breaking region was identified by showing the linear instability of
the radial extremals. Here we prove that symmetry can be broken even within the
set of parameters where radial extremals correspond to local minima for the
variational problem associated with the inequality. For interpolation
inequalities, such a symmetry breaking phenomenon is entirely new.

Analysis of PDEs