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arXiv · 2310.01359

On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations

Changxing Miao·Zhiwen Zhao
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Abstract

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.

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Changxing Miao, Zhiwen Zhao. 2024-11-18. On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations. https://arxiv.org/abs/2310.01359

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