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Regularity of monotone transport maps between unbounded domains

  • * Corresponding author: Alessio Figalli

    * Corresponding author: Alessio Figalli

The second author has received funding from the European Research Council under the Grant Agreement No. 721675 "Regularity and Stability in Partial Differential Equations (RSPDE)"

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  • The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we show that, in several situations of interest, one can to ensure the regularity of monotone maps even if the measures may have unbounded supports.

    Mathematics Subject Classification: Primary: 35J60, 49Q20; Secondary: 35J96, 49N60.


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  • Figure 1.  We subtract the affine function $ \ell_\varepsilon(z): = \varepsilon(z_1+\eta_0) $ from $ u $. Because $ \mathbf{0} \in \partial ({\rm dom}(u)) $, $ u_\varepsilon|_{\partial S_\varepsilon} $ contains some vertical segments in its graph

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