Poincaré's inequality and Sobolev spaces with monomial weights

Hernán Castro and Marco Cornejo

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4500-4522

Abstract: In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set Ω⊆RN$\Omega \subseteq \mathbb {R}^N$. Additionally, we study the specific case of monomial weights w(x1,…,xN)=∏i=1Nxiai,ai≥0$w(x_1,\ldots ,x_N)=\prod _{i=1}^N\left|x_i \right|^{a_i},\ a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.

Date: 2023
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https://doi.org/10.1002/mana.202200100

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