Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument
The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Mose...
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| Format: | Article |
| Language: | English |
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De Gruyter
2021-11-01
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| Series: | Advanced Nonlinear Studies |
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| Online Access: | https://doi.org/10.1515/ans-2021-2147 |
| _version_ | 1817997001257648128 |
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| author | Li Jungang Lu Guozhen Zhu Maochun |
| author_facet | Li Jungang Lu Guozhen Zhu Maochun |
| author_sort | Li Jungang |
| collection | DOAJ |
| description | The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold. |
| first_indexed | 2024-04-14T02:31:48Z |
| format | Article |
| id | doaj.art-1580571396f1419abe1fc5a9725a44ba |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-14T02:31:48Z |
| publishDate | 2021-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-1580571396f1419abe1fc5a9725a44ba2022-12-22T02:17:40ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752021-11-0121491793710.1515/ans-2021-2147Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free ArgumentLi Jungang0Lu Guozhen1Zhu Maochun2Department of Mathematics, Brown University, Providence, RI 02912, USADepartment of Mathematics, University of Connecticut, Storrs, CT 06269, USASchool of Mathematical Sciences, Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, P. R. ChinaThe concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.https://doi.org/10.1515/ans-2021-2147trudinger–moser inequalityheisenberg groupconcentration-compactness principlesriemannian manifoldsexponential growth46e35 53c21 42b35 |
| spellingShingle | Li Jungang Lu Guozhen Zhu Maochun Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument Advanced Nonlinear Studies trudinger–moser inequality heisenberg group concentration-compactness principles riemannian manifolds exponential growth 46e35 53c21 42b35 |
| title | Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument |
| title_full | Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument |
| title_fullStr | Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument |
| title_full_unstemmed | Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument |
| title_short | Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument |
| title_sort | concentration compactness principle for trudinger moser s inequalities on riemannian manifolds and heisenberg groups a completely symmetrization free argument |
| topic | trudinger–moser inequality heisenberg group concentration-compactness principles riemannian manifolds exponential growth 46e35 53c21 42b35 |
| url | https://doi.org/10.1515/ans-2021-2147 |
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