DOI 10.1007/s10958-018-3879-9 Journal of Mathematical Sciences, Vol. 232, No. 4, July, 2018 A COUNTEREXAMPLE RELATED TO THE REGULARITY OF THE p-STOKES PROBLEM M. Kˇrepela Institute of Applied Mathematics, University of Freiburg Eckerstr. 1, D-79104 Freiburg, Germany email@example.com M. R˚ uˇziˇcka Institute of Applied Mathematics, University of Freiburg Eckerstr. 1, D-79104 Freiburg, Germany firstname.lastname@example.org UDC 517.9 1,s 2,q We construct a solenoidal vector ﬁeld u belonging to W (Ω) ∩ W (Ω), q ∈ (1,n), s ∈ p−2 (1, ∞), such that (1 + |Du|) , 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. Bibliography:28 titles. Dedicated to V. V. Zhikov 1 Introduction The question of full natural regularity for the p-Stokes system − div S(Du)+ ∇π = f in Ω, (1.1) div u =0 in Ω, is still not completely solved in the three- and higher-dimensional situation. Here, Ω ⊂ R , n 2, is a bounded domain. The unknowns are the velocity vector ﬁeld u =(u ,...,u ) and 1 n the scalar pressure π, while the external body force
Journal of Mathematical Sciences – Springer Journals
Published: Jun 5, 2018
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