Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents
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| Format: | Preprint |
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2021
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| _version_ | 1866929244509044736 |
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| author | Du, Shi-Zhong Liu, YanNan Lu, Jian |
| author_facet | Du, Shi-Zhong Liu, YanNan Lu, Jian |
| contents | In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Ampère type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-λ_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $λ_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincaré inequality on $\mathbb{S}^{n-1}$ with k-symmetricity. The second part of this paper is devoted to prove some new nonexistence results for the supercritical range $\mathit{p}\leq-\mathit{q}, \mathit{q}\geq\mathit{n}$ on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for $\mathit{q}=\mathit{n}$, $\mathit{p}$=$-\mathit{q}$ to a full range of $(\mathit{p},\mathit{q})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_07426 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents Du, Shi-Zhong Liu, YanNan Lu, Jian Analysis of PDEs 35J20, 35J60, 52A40, 53A15 In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Ampère type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-λ_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $λ_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincaré inequality on $\mathbb{S}^{n-1}$ with k-symmetricity. The second part of this paper is devoted to prove some new nonexistence results for the supercritical range $\mathit{p}\leq-\mathit{q}, \mathit{q}\geq\mathit{n}$ on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for $\mathit{q}=\mathit{n}$, $\mathit{p}$=$-\mathit{q}$ to a full range of $(\mathit{p},\mathit{q})$. |
| title | Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents |
| topic | Analysis of PDEs 35J20, 35J60, 52A40, 53A15 |
| url | https://arxiv.org/abs/2104.07426 |