About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

Jean Dolbeault (1) , Maria J. Esteban (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Mots clés: Caffarelli-Kohn-Nirenberg inequality; Gagliardo-Nirenberg inequality; logarithmic Hardy inequality; logarithmic Sobolev inequality; extremal functions; radial symmetry; symmetry breaking; Emden-Fowler transformation; linearization; existence; compactness; optimal constants

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban. About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities. The Abel Symposium 2010, 2010, Norway. pp.117-130. ⟨hal-00539019⟩