Abstract
We study the solvability of the divergence equation in weighted spaces and Lebesgue spaces with variable exponents, where the weights are so called Muckenhoupt weights. The question of constructing divergence free test functions, which can be used for problems arising in fluid dynamics, is also addressed. The approach is based on an explicit representation formula for solutions of the divergence equation due to Bogovskiĭ and the theory of singular integral operators. The developed methods are used to prove an existence result for fluids which satisfy a p(·)-growth condition.
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Acknowledgments
This work is a short version of the author’s diploma thesis submitted in September 2005 at the University of Freiburg, Germany. The author would like to thank his supervisor Michael Růžička for suggesting the topic and continuous support. In addition the author would like to thank Lars Diening for stimulating discussions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Huber, A. The divergence equation in weighted- and L p(·)-spaces. Math. Z. 267, 341–366 (2011). https://doi.org/10.1007/s00209-009-0622-8
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DOI: https://doi.org/10.1007/s00209-009-0622-8
Keywords
- Divergence equation
- Muckenhoupt weights
- Lebesgue spaces with variable exponents
- Singular integrals
- Fluids with p(·)-growth