Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces
Dolbeault, Jean; Esteban, Maria J.; Loss, Michael (2016), Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Inventiones Mathematicae, 206, 2, p. 397-440. 10.1007/s00222-016-0656-6
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/1506.03664v1
Journal nameInventiones Mathematicae
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Esteban, Maria J.
Abstract (EN)This paper is motivated by the characterization of the optimal symmetry breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence, optimal functions and sharp constants are computed in the symmetry region. The result solves a longstanding conjecture on the optimal symmetry range. As a byproduct of our method we obtain sharp estimates for the principal eigenvalue of Schrödinger operators on some non-flat non-compact manifolds, which to the best of our knowledge are new. The method relies on generalized entropy functionals for nonlinear diffusion equations. It opens a new area of research for approaches related to carré du champ methods on non-compact manifolds. However, key estimates depend as much on curvature properties as on purely nonlinear effects. The method is well adapted to functional inequalities involving simple weights and also applies to general cylinders. Beyond results on symmetry and symmetry breaking, and on optimal constants in functional inequalities, rigidity theorems for nonlinear elliptic equations can be deduced in rather general settings.
Subjects / KeywordsCaffarelli-Kohn-Nirenberg inequalities; symmetry; symmetry breaking; optimal constants; rigidity results; fast diffusion equation; carré du champ; bifurcation; instability; Emden-Fowler transformation; cylinders; non-compact manifolds; Laplace-Beltrami operator; spectral estimates; Keller-Lieb-Thirring estimate; Hardy inequality
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