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Jean Dolbeault

Symmetrization based on entropy methods and nonlinear flows

Abstract: Some Gagliardo-Nirenberg inequalities are equivalent to entropy - entropy production inequalities and can be established by the Bakry-Emery, or carré du champ, method applied to fast diffusion equations. In presence of weights, adapted nonlinear flows can also be used as tools to study symmetry and symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities. Large time asymptotics determine which regime of symmetry has to be considered. This lecture will emphasize the role of the linearized inequalities in the study of the nonlinear evolution equations and the consequences for nonlinear interpolation inequalities and the determination of the corresponding optimal constants.