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Geometric PDE - Optimal transportation and nonlinear elliptic PDE

Neil Trudinger
Institute for Advanced Study
November 11, 2008

http://www.math.ias.edu/sp/gpde

 

In these lectures we will describe the relationship between optimal transportation and nonlinear elliptic PDE of Monge-Ampere type, focusing on recent advances in characterizing costs and domains for which the Monge-Kantorovich problem has smooth diffeomorphism solutions.

Background references.

L.C Evans, PDE and Monge-Kantorovich mass transfer. Current developments in Mathematics, 1997. Int. Press, Boston, (1999).

Geometric PDE - Variational techniques for the prescribed Q-curvature equation

Andrea Malchiodi
Institute for Advanced Study
October 21, 2008

http://www.math.ias.edu/sp/gpde

 

After recalling the definition of Q-curvature and some applications, we will address the question of prescribing it through a conformal deformation of the metric. We will address some compactness issues, treated via blow-up analysis, and then study the problem, which has variational structure, using a Morse-theoretical approach.