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Partial Regularity in the Second Boundary Value Problem for Generated Jacobian Equations

Banff International Research Station for Mathematical Innovation and Discovery

Full regularity results for solutions to the SBVP for GJEs require strong geometric conditions on the domains of the problem as well as higher order structure conditions on the generating function. For example, in the optimal transport problem, these are $c$-convexity and $c^*$-convexity restrictions on the source and target domains respectively and the MTW conditions. We extend the partial regularity result of De Philippis and Figalli from the optimal transport setting to the general GJE setting and show that without any geometric conditions on the domains or additional structure conditions, akin to the MTW conditions, on the generating function, solutions are smooth outside a closed singular set of measure zero. This result is especially relevant to the general GJE framework when applied to problems in geometric optics: in the reflector shape design problem, Karakhanyan and Wang show that smooth data may yield many solutions each with different regularity properties.

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

video/mp4

Author affiliation: ETH Zurich

2017-10-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63217

Unreviewed

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