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Radial symmetry and symmetry breaking for some interpolation inequalities. (English) Zbl 1246.26014

The symmetric properties of extremals for a class of interpolation inequalities known as Caffarelli-Kohn-Nirenberg inequalities are established. Furthermore, the class of logarithmic Hardy type inequalities obtained as limiting cases of Caffarelli-Kohn-Nirenberg inequalities are derived. For these classes of inequalities, it was shown that there exists a continuous surface that splits the set of admissible parameters into a region where extremals are symmetric and a region where symmetric breaking occurs. It was also proved that symmetry can be broken even within the set of parameters where radial extremals correspond to local minima for the variational problems associated with the inequality. In addition, it was shown that for interpolation inequalities such a symmetry breaking phenomenon is an entirely new concept. Finally, it was also shown that the weighted logarithmic Hardy inequality when compared with the logarithm Sobolev inequality gives a better information about the symmetry breaking properties of the extremals.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J40 Variational inequalities
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