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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2310.01359 |
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| _version_ | 1866916483951493120 |
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| author | Miao, Changxing Zhao, Zhiwen |
| author_facet | Miao, Changxing Zhao, Zhiwen |
| contents | In this paper, a class of anisotropic weights having the form of $|x'|^{θ_{1}}|x|^{θ_{2}}|x_{n}|^{θ_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(θ_{1},θ_{2},θ_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{p}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{p}$. As a consequence, we obtain anisotropic weighted Poincaré and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01359 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations Miao, Changxing Zhao, Zhiwen Analysis of PDEs In this paper, a class of anisotropic weights having the form of $|x'|^{θ_{1}}|x|^{θ_{2}}|x_{n}|^{θ_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(θ_{1},θ_{2},θ_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{p}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{p}$. As a consequence, we obtain anisotropic weighted Poincaré and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations. |
| title | On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.01359 |