Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces

• 成果类型: Article
• 署名作者: Dolbeault, Jean; Esteban, Maria J.; Loss, Michael
• 署名单位: Universite PSL; Universite Paris-Dauphine; University System of Georgia; Georgia Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0656-6
• 发表日期: 2016
• 页码: 397-440
• 关键词: kohn-nirenberg inequalities; compact riemannian-manifolds; hardy-littlewood-sobolev; elliptic-equations; interpolation inequalities; extremal-functions; sharp constants; asymptotics; BEHAVIOR; weights

摘要

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 Schrodinger 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 carr, 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.