Area minimizing discs in locally non-compact metric spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866913543648968704 |
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| author | Guo, Chang-Yu Wenger, Stefan |
| author_facet | Guo, Chang-Yu Wenger, Stefan |
| contents | We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1701_06736 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Area minimizing discs in locally non-compact metric spaces Guo, Chang-Yu Wenger, Stefan Metric Geometry Analysis of PDEs 49Q05, 53C23 We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps. |
| title | Area minimizing discs in locally non-compact metric spaces |
| topic | Metric Geometry Analysis of PDEs 49Q05, 53C23 |
| url | https://arxiv.org/abs/1701.06736 |