Abstract
We collect some examples of optimal transports for quadratic cost in order to explore the stability of the identity map as an optimal transport. First, we consider density and domain perturbations near regular portions of domains. Second, we investigate density and domains deformations to non-regular parts of domains. Here, we restrict our attention to two dimensions and focus near 90\(^{\circ }\) corners.
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Notes
In actuality, he showed that the optimal transport from \(B_1\) to a smoothing of \(D_\varepsilon \) is discontinuous. However, the regularity of \(D_\varepsilon \) is irrelevant to the essence of the singular nature of his example.
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Acknowledgements
I would like to thank Connor Mooney for some keen observations and suggestions, particularly with respect to the barrier argument in the proof of Theorem 2.2. Also, I’m grateful to Alessio Figalli for his encouragement and guidance. YJ was supported in part by the ERC grant “Regularity and Stability in Partial Differential Equations (RSPDE)”.
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Communicated by N. Trudinger.
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Jhaveri, Y. On the (in)stability of the identity map in optimal transportation. Calc. Var. 58, 96 (2019). https://doi.org/10.1007/s00526-019-1536-x
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DOI: https://doi.org/10.1007/s00526-019-1536-x