Radial symmetry and symmetry breaking for some interpolation inequalities
Tertikas, Achilles; Tarantello, Gabriella; Esteban, Maria J.; Dolbeault, Jean (2011), Radial symmetry and symmetry breaking for some interpolation inequalities, Calculus of Variations and Partial Differential Equations, 42, 3-4, p. 461-485. http://dx.doi.org/10.1007/s00526-011-0394-y
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00516710/fr/
Journal nameCalculus of Variations and Partial Differential Equations
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Abstract (EN)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.
Subjects / Keywordslogarithmic Sobolev inequality; Kelvin transformation; compactness; existence; linearization; symmetry breaking; radial symmetry; Emden-Fowler transformation; scale invariance; Gagliardo-Nirenberg inequality; logarithmic Hardy inequality; Caffarelli-Kohn-Nirenberg inequality; Hardy-Sobolev inequality; interpolation; Sobolev spaces; extremal functions
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