Formulas for the $h$-mass on $1$-currents with coefficients in $\mathbb{R}^m$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Lohmann, Julius Schmitzer, Bernhard Wirth, Benedikt |
| author_facet | Lohmann, Julius Schmitzer, Bernhard Wirth, Benedikt |
| contents | We consider the minimization of the $h$-mass over normal $1$-currents in $\mathbb{R}^n$ with coefficients in $\mathbb{R}^m$ and prescribed boundary. This optimization is known as multi-material transport problem and used in the context of logistics of multiple commodities, but also as a relaxation of nonconvex optimal transport tasks such as so-called branched transport problems. The $h$-mass with norm $h$ can be defined in different ways, resulting in three functionals $\mathcal{M}_h,|\cdot|_H$, and $\mathbb{M}_h$, whose equality is the main result of this article: $\mathcal{M}_h$ is a functional on $1$-currents in the spirit of Federer and Fleming, norm $|\cdot|_H$ denotes the total variation of a Radon measure with respect to $H$ induced by $h$, and $\mathbb{M}_h$ is a mass on flat $1$-chains in the sense of Whitney. On top we introduce a new and improved notion of calibrations for the multi-material transport problem: we identify calibrations with (weak) Jacobians of optimizers of the associated convex dual problem, which yields their existence and natural regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formulas for the $h$-mass on $1$-currents with coefficients in $\mathbb{R}^m$ Lohmann, Julius Schmitzer, Bernhard Wirth, Benedikt Optimization and Control We consider the minimization of the $h$-mass over normal $1$-currents in $\mathbb{R}^n$ with coefficients in $\mathbb{R}^m$ and prescribed boundary. This optimization is known as multi-material transport problem and used in the context of logistics of multiple commodities, but also as a relaxation of nonconvex optimal transport tasks such as so-called branched transport problems. The $h$-mass with norm $h$ can be defined in different ways, resulting in three functionals $\mathcal{M}_h,|\cdot|_H$, and $\mathbb{M}_h$, whose equality is the main result of this article: $\mathcal{M}_h$ is a functional on $1$-currents in the spirit of Federer and Fleming, norm $|\cdot|_H$ denotes the total variation of a Radon measure with respect to $H$ induced by $h$, and $\mathbb{M}_h$ is a mass on flat $1$-chains in the sense of Whitney. On top we introduce a new and improved notion of calibrations for the multi-material transport problem: we identify calibrations with (weak) Jacobians of optimizers of the associated convex dual problem, which yields their existence and natural regularity. |
| title | Formulas for the $h$-mass on $1$-currents with coefficients in $\mathbb{R}^m$ |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2407.10158 |