Transport of currents and geometric Rademacher-type theorems
speaker: Paolo Bonicatto (University of Warwick)abstract: In the classical theory, given a vector field $b$ on $\mathbb Rd$, one usually studies the transportcontinuity equation drifted by $b$ looking for solutions in the class of functions (with certain integrability) or at most in the class of measures. In this seminar I will talk about recent efforts, motivated by the modeling of defects in plastic materials, aimed at extending the previous theory to the case when the unknown is instead a family of k-currents in $\mathbb Rd$, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivative $Lb$ of currents in direction $b$ and reads $\partialt Tt + Lb Tt = 0$. In the first part of the talk I will briefly introduce this equation, with a special attention to its space-time formulation. I will then shift the focus to some rectifiability questions and Rademacher-type results: given a Lipschitz path of integral currents, I will discuss the existence of a “geometric derivative”, namely a vector field advecting the currents. Based on joint work with G. Del Nin and F. Rindler (Warwick).
timetable:
Wed 14 Sep, 11:15 - 12:15, Aula Dini
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