Hardy-Poincaré inequalities and applications to nonlinear diffusions

Adrien Blanchet (1) , Matteo Bonforte (1) , Jean Dolbeault (1) , Gabriele Grillo (2) , Juan-Luis Vázquez (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Dipartimento di Matematica "Giuseppe Peano" [Torino]
3 Departamento de Matemáticas [Madrid]

We systematically study weighted Poincaré type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics results for fast diffusion equations.

Keywords: Poincaré inequalities; Hardy inequalities; weighted Sobolev spaces; fast diffusion equations; selfsimilar solutions; large time asymptotics; asymptotics near extinction time; entropy methods

Domains: Analysis of PDEs [math.AP]

Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo, Juan-Luis Vázquez. Hardy-Poincaré inequalities and applications to nonlinear diffusions. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2007, 344 (7), pp.431-436. ⟨hal-00105661⟩