Optimal concentration level of anisotropic Trudinger-Moser functionals on any bounded domain
| Autor: | Chen, Lu, Jiang, Rou, Zhu, Maochun |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $F$ be convex and homogeneous of degree $1$, its polar $F^{o}$ represent a finsler metric on $\mathbb{R}^{n}$, and $\Omega$ be any bounded open set in $\mathbb{R}^{n}$. In this paper, we first construct the theoretical structure of anisotropic harmonic transplantation. Using the anisotropic harmonic transplantation, co-area formula, limiting Sobolev approximation method, delicate estimate of level set of Green function, we investigate the optimal concentration level of the Trudinger-Moser functional \[ \int_{\Omega}e^{\lambda_{n}|u|^{\frac{n}{n-1}}}dx \] under the anisotropic Dirichlet norm constraint $\int_{\Omega}F^{n}\left( \nabla{{u}}\right) dx\leq1$, where $\lambda_{n}=n^{\frac{n}{n-1}}\kappa _{n}^{\frac{1}{n-1}}\ $ denotes the sharp constant of anisotropic Trudinger-Moser inequality in bounded domain and $\kappa_{n}$ is the Lebesgue measure of the unit Wulff ball. As an application. we can immediately deduce the existence of extremals for anisotropic Trudinger-Moser inequality on bounded domain. Finally, we also consider the optimal concentration level of the anisotropic singular Trudinger-Moser functional. The method is based on the limiting Hardy-Sobolev approximation method and constructing a suitable normalized anisotropic concentrating sequence. Comment: When using the polynomial approximation functional to prove the optimal concentration upper bound of the Trudinger-Moser inequality, we can not show that the order of limits can be exchanged |
| Databáze: | arXiv |
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