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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.09051 |
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| _version_ | 1866916593868472320 |
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| author | Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru |
| author_facet | Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru |
| contents | We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted $p$-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_09051 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru Analysis of PDEs We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted $p$-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants. |
| title | Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2205.09051 |