Symmetry and Nonlinear Diffusion Flows

These notes are the lecture notes of the course Symmetry and nonlinear diffusion flows http://verso.mat.uam.es/∼difusion.nolineal/pdes-geometry-lectures/ Lectures on PDEs and Geometry Universidad Autónoma de Madrid (Spain) The material has been collected from various papers and publications. The goal is to provide additional details on the proofs and references, without covering the most general cases. Topics covered are: • a summary of known results on φ-entropies and related functional inequalities based on simple linear diffusion equations, including results based on the Bakry-Emery method, • interpolation inequalities on compact manifolds, with the sphere as main example, with an emphasis on the use of the fast diffusion flow in order to cover the whole range of parameters up to the critical exponent, • Rényi entropy powers compared to relative entropy methods on the Euclidean space as a new tool for capturing optimal constants in the large time regime, with applications to Gagliardo-Nirenberg inequalities, • Considerations on branches of solutions and bifurcations in semilinear elliptic equations: known rigidity results can be reinterpreted as stationary points of flows based on nonlinear diffusions, • Symmetry and symmetry breaking results in Caffarelli-Kohn-Nirenberg inequalities: how to introduce nonlinear flows in presence of weights for proving symmetry reults, • Further considerations on large time asymptotics, linearization and optimal constants. CONTENTS 1. A review of results on φ-entropies 1 1.1. Generalized Csiszár-Kullback-Pinsker inequality 3 1.2. Convexity, tensorization and sub-additivity 3 1.3. Entropy – entropy production inequalities: perturbation results 6 1.4. Entropy – entropy production inequalities and linear flows 6 1.5. Improved entropy – entropy production inequalities 8 2. The carré du champ method on the sphere 9 3. Symmetry and symmetry breaking results in critical Caffarelli-Kohn-Nirenberg inequalities 9 3.