Seminar Analysis and Mathematical Physics: Paolo Bonicatto (University of Warwick)

Transport of currents and geometric Rademacher-type theorems

In the classical theory, given a vector field $b$ on $\mathbb R^d$, one usually studies the transport/continuity 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 modelling 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 $\mathbb R^d$, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivative $L_b$ of currents in direction $b$ and reads $\partial_t T_t + L_b T_t = 0$. In the first part of the talk I will briefly introduce this equation, with 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. Joint work with G. Del Nin and F. Rindler (Warwick).

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