Uniqueness of Lp Minkowski problem in the supercritical range
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911262778064896 |
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| author | Du, Shi-Zhong |
| author_facet | Du, Shi-Zhong |
| contents | The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents $p\in(-n-1,1]$. In the supercritical range, the uniqueness problem is much more complicated, even on the planar case $n=1$. One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range $p\in(-7,-2)$ and fails to hold for the other supercritical exponents $p\in(-\infty,-7)$. In this paper, we study the same uniqueness problem in the full supercritical range $p\in(-2n-5,-n-1)$ for all higher dimensional cases $n\geq2$. We will prove that for $p\in(-2n-5,-n-1)$, the unique strongly symmetric solution is given by the unit sphere $§^n$. The uniqueness range $(-2n-5,-n-1)$ is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all $p\in(-\infty,-2n-5)$. When considering general solutions which may not be symmetric, the uniqueness set $Γ$ of $p$ for which the uniqueness holds, is shown to be both relatively open and closed in the full interval $(-2n-5,-n-1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06946 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of Lp Minkowski problem in the supercritical range Du, Shi-Zhong Analysis of PDEs The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents $p\in(-n-1,1]$. In the supercritical range, the uniqueness problem is much more complicated, even on the planar case $n=1$. One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range $p\in(-7,-2)$ and fails to hold for the other supercritical exponents $p\in(-\infty,-7)$. In this paper, we study the same uniqueness problem in the full supercritical range $p\in(-2n-5,-n-1)$ for all higher dimensional cases $n\geq2$. We will prove that for $p\in(-2n-5,-n-1)$, the unique strongly symmetric solution is given by the unit sphere $§^n$. The uniqueness range $(-2n-5,-n-1)$ is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all $p\in(-\infty,-2n-5)$. When considering general solutions which may not be symmetric, the uniqueness set $Γ$ of $p$ for which the uniqueness holds, is shown to be both relatively open and closed in the full interval $(-2n-5,-n-1)$. |
| title | Uniqueness of Lp Minkowski problem in the supercritical range |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.06946 |