A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure
In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a M...
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| Format: | Article |
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AIMS Press
2023-01-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://aimspress.com/article/doi/10.3934/era.2023042?viewType=HTML |
| _version_ | 1811168541753212928 |
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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 existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor. |
| first_indexed | 2024-04-10T16:27:53Z |
| format | Article |
| id | doaj.art-505a3939e3604ce9af68760ec447e59d |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-10T16:27:53Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-505a3939e3604ce9af68760ec447e59d2023-02-09T01:12:44ZengAIMS PressElectronic Research Archive2688-15942023-01-0131284085910.3934/era.2023042A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measureZhengmao Chen0School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, ChinaIn the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.https://aimspress.com/article/doi/10.3934/era.2023042?viewType=HTMLcodazzi tensorlog-convex measuresl<sub>p</sub> aleksandrov problemthe continuous methoda priori bounds |
| spellingShingle | Zhengmao Chen A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure Electronic Research Archive codazzi tensor log-convex measures l<sub>p</sub> aleksandrov problem the continuous method a priori bounds |
| title | A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure |
| title_full | A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure |
| title_fullStr | A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure |
| title_full_unstemmed | A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure |
| title_short | A priori bounds and existence of smooth solutions to a Lp Aleksandrov problem for Codazzi tensor with log-convex measure |
| title_sort | priori bounds and existence of smooth solutions to a lp aleksandrov problem for codazzi tensor with log convex measure |
| topic | codazzi tensor log-convex measures l<sub>p</sub> aleksandrov problem the continuous method a priori bounds |
| url | https://aimspress.com/article/doi/10.3934/era.2023042?viewType=HTML |
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