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arXiv · 2409.19127

Differentiability of monotone maps related to non-quadratic costs

Cristian E. Gutiérrez·Annamaria Montanari
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Abstract

The cost functions considered are $c(x,y)=h(x-y)$, with $h\in C^2(R^n)$, homogeneous of degree $p\geq 2$, with positive definite Hessian in the unit sphere. We consider monotone maps $T$ concerning that cost and establish local $L^\infty$-estimates of $T$ minus affine functions, which are applied to obtain differentiability properties of $T$ a.e. It is also shown that these maps are related to maps of bounded deformation, and further, differentiability and H\"older continuity properties are derived.

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Cristian E. Gutiérrez, Annamaria Montanari. 2024-09-27. Differentiability of monotone maps related to non-quadratic costs. https://arxiv.org/abs/2409.19127

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