Metric Sobolev spaces II: dual energies and divergence measures
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917012055261184 |
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| author | Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico |
| author_facet | Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico |
| contents | This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents $p\in[1,\infty)$ and without finiteness assumption on the measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric Sobolev spaces II: dual energies and divergence measures Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico Functional Analysis Analysis of PDEs Metric Geometry 49J52, 46E35, 53C23, 46N10, 46E15, 31C15, 28A12 This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents $p\in[1,\infty)$ and without finiteness assumption on the measure. |
| title | Metric Sobolev spaces II: dual energies and divergence measures |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 49J52, 46E35, 53C23, 46N10, 46E15, 31C15, 28A12 |
| url | https://arxiv.org/abs/2510.12424 |