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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.02806 |
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| _version_ | 1866910639454158848 |
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| author | Candau-Tilh, Jules Goldman, Michael Merlet, Benoît |
| author_facet | Candau-Tilh, Jules Goldman, Michael Merlet, Benoît |
| contents | This paper deals with a variant of the optimal transportation problem. Given f $\in$ L 1 (R d , [0, 1]) and a cost function c $\in$ C(R d x R d) of the form c(x, y) = k(y -- x), we minimise $\int$ c d$γ$ among transport plans $γ$ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 -- f. Denoting by $Υ$(f) the infimum of this problem, we then consider the maximisation problem sup{$Υ$(f) : $\int$ f = m} where m \> 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02806 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An exterior optimal transport problem Candau-Tilh, Jules Goldman, Michael Merlet, Benoît Analysis of PDEs This paper deals with a variant of the optimal transportation problem. Given f $\in$ L 1 (R d , [0, 1]) and a cost function c $\in$ C(R d x R d) of the form c(x, y) = k(y -- x), we minimise $\int$ c d$γ$ among transport plans $γ$ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 -- f. Denoting by $Υ$(f) the infimum of this problem, we then consider the maximisation problem sup{$Υ$(f) : $\int$ f = m} where m \> 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m. |
| title | An exterior optimal transport problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2309.02806 |