Seminar Analysis: Emil Wiedemann (University of British Columbia)

Incompressible Extensions in Sobolev Spaces

Given a bounded domain and boundary data, does there exist a vector-valued map on this domain which is incompressible, that is, a map whose Jacobian determinant is one (almost) everywhere? In a regular setting, this question has been essentially positively answered in a famous paper by Dacorogna and Moser. I will present an analogous result in Sobolev spaces of low regularity, which was recently achieved by a convex integration method jointly with K. Koumatos (Oxford) and F. Rindler (Warwick). I will also comment on several generalisations and applications.

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