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Liftings of BV-maps and lower semicontinuity

Banff International Research Station for Mathematical Innovation and Discovery

Liftings and their associated Young measures are new tools to study the asymptotic behaviour of sequences of BV-maps under weak* convergence. Their main feature is that they allow to keep track of the precise shape of the jump path and as such are natural objects whenever different ways of approaching a jump need to be distinguished. While this tool has several promising applications, in this talk I will focus on its use to prove lower semicontinuity for linear-growth functionals that depend on the value of the argument function, u(x), besides its gradient. It is well known that in this situation the particular shape of jumps cannot be neglected. Using the theory of liftings, we can prove relaxation theorems under essentially optimal assumptions, generalizing a classical theorem by Fonseca & Müller (1993). The key idea is that liftings provide the right way of localizing the functional in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow up procedure. This is joint work with Giles Shaw.

Mathematics; Calculus of variations and optimal control; optimization; Partial differential equations

video/mp4

Author affiliation: University of Warwick

2019-03-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69009

Unreviewed

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