A trajectorial interpretation of Moser's proof of the Harnack inequality
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917028318674944 |
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| author | Niebel, Lukas Zacher, Rico |
| author_facet | Niebel, Lukas Zacher, Rico |
| contents | In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincaré inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07977 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A trajectorial interpretation of Moser's proof of the Harnack inequality Niebel, Lukas Zacher, Rico Analysis of PDEs 35K10, 35B45, 35B65, 35J15 In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincaré inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations. |
| title | A trajectorial interpretation of Moser's proof of the Harnack inequality |
| topic | Analysis of PDEs 35K10, 35B45, 35B65, 35J15 |
| url | https://arxiv.org/abs/2212.07977 |