A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms
In the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $ e^{-f(|x|^2)}dx $ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the anal...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-04-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023663?viewType=HTML |
| _version_ | 1797848244739375104 |
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| author | Zhengmao Chen |
| author_facet | Zhengmao Chen |
| author_sort | Zhengmao Chen |
| collection | DOAJ |
| description | In the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $ e^{-f(|x|^2)}dx $ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the analysis is the a priori bounds of an auxiliary Monge-Ampère equation on $ \mathbb{S}^n $. The main result of the present paper extends the Minkowski type problem of log-concave measures to the space forms and it may be an attempt to get some new analysis for the log-concave measures. |
| first_indexed | 2024-04-09T18:24:23Z |
| format | Article |
| id | doaj.art-782c791f1b5a490aab2a267f60ebd155 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T18:24:23Z |
| publishDate | 2023-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-782c791f1b5a490aab2a267f60ebd1552023-04-12T01:26:44ZengAIMS PressAIMS Mathematics2473-69882023-04-0186131341315310.3934/math.2023663A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space formsZhengmao Chen 0School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, ChinaIn the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $ e^{-f(|x|^2)}dx $ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the analysis is the a priori bounds of an auxiliary Monge-Ampère equation on $ \mathbb{S}^n $. The main result of the present paper extends the Minkowski type problem of log-concave measures to the space forms and it may be an attempt to get some new analysis for the log-concave measures.https://www.aimspress.com/article/doi/10.3934/math.2023663?viewType=HTMLlog-concave measureminkowski problemmonge-ampère equationthe continuous methodwarped product space forms |
| spellingShingle | Zhengmao Chen A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms AIMS Mathematics log-concave measure minkowski problem monge-ampère equation the continuous method warped product space forms |
| title | A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms |
| title_full | A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms |
| title_fullStr | A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms |
| title_full_unstemmed | A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms |
| title_short | A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms |
| title_sort | priori bounds and existence of smooth solutions to minkowski problems for log concave measures in warped product space forms |
| topic | log-concave measure minkowski problem monge-ampère equation the continuous method warped product space forms |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023663?viewType=HTML |
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