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Uniqueness and symmetry based on nonlinear flows

Banff International Research Station for Mathematical Innovation and Discovery

The \emph{carr\'e du champ} method and its nonlinear counterpart is a powerful technique to prove uniqueness in some nonlinear elliptic PDEs and establish optimal constants in interpolation inequalities or optimal rates of decay in related evolution problems. It is related with entropy methods in PDEs inspired by generalizations of Boltzmann's entropy. Optimality cases can be identified by considering asymptotic regimes and appropriate linearizations. The method applies to symmetry results in presence of weights. It does not rely on symmetrization but raises various issues of regularity and rely on integrations by parts which are not always straightforward to justify. This lecture will be devoted to an overview of the main results.

Mathematics; Partial differential equations; Global analysis, analysis on manifolds

video/mp4

Author affiliation: Université Paris-Dauphine

2018-10-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67293

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/