We construct a solenoidal vector field u belonging to \( {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right),q\in \left(1,n\right),s\in \left(1,\infty \right) \), such that (1 + |Du|)p − 2, p ∈ (1, ∞), p ≠ 2, does not belong to the Muckenhoupt class A∞(Ω). Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the p-Stokes problem.
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Dedicated to V. V. Zhikov
Translated from Problemy Matematicheskogo Analiza 92, 2018, pp. 159-168.
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Křepela, M., Růžička, M. A Counterexample Related to the Regularity of the p-Stokes Problem. J Math Sci 232, 390–401 (2018). https://doi.org/10.1007/s10958-018-3879-9
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DOI: https://doi.org/10.1007/s10958-018-3879-9