New Trends in Quantum and Classical Kinetic Equations and Related PDEs

Paths of minimal lengths on the set of exact differential k–forms

Wilfrid Gangbo

Georgia Institute of Technology

We initiate the study of optimal transportation of exact differential k–forms and introduce various distances as minimal actions. Our study involves dual maximization problems with constraints on the codifferential of k–forms. When k < n, only some direc- tional derivatives of a vector field are controlled. This is in contrast with prior studies of optimal transportation of volume forms (k = n), where the full gradient of a scalar function is controlled. Furthermore, our study involves paths of bounded variations on the set of k–currents. This talk is based a joint work with B. Dacorogna and O. Kneuss.