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Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension
- Publication Year :
- 2022
-
Abstract
- We study certain inequalities and a related result on weighted Sobolev spaces on bounded John domains in $\mathbb{R}^n$. Namely, we prove the existence of a right inverse for the divergence operator, along with the corresponding a priori estimate, the improved and the fractional Poincar\'e inequalities, the Korn inequality and the local Fefferman-Stein inequality. All these results are obtained on weighted Sobolev spaces, where the weight is a power of the distance to the boundary. In all cases the exponent of the weight $d(\cdot,\partial\Omega)^{\beta p}$ is only required to satisfy the restriction: $\beta p>-(n-\dim_A(\partial\Omega))$, where $p$ is the exponent of the Sobolev space and $\dim_A(\partial\Omega)$ is the Assouad dimension of the boundary of the domain. According to our best knowledge, this condition is less restrictive than the ones in the literature.<br />Comment: 21 pages, 2 figures
- Subjects :
- Mathematics - Analysis of PDEs
Primary: 26D10, Secondary: 31E05, 35A23, 46E35
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2210.12322
- Document Type :
- Working Paper