Higher differentiability for solutions of stationary p‐Stokes systems

We consider weak solutions (u,π):Ω→Rn×R to stationary p‐Stokes systems of the type −diva(x,Eu)+∇π+[Du]u=f,divu=0in Ω⊂Rn, where the function a(x,ξ) satisfies p‐growth conditions in ξ and depends Hölder continuously on x. By Eu we denote the symmetric part of the gradient Du and we write [Du]u for the...

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Bibliographic Details
Published in:Mathematische Nachrichten 2020-11, Vol.293 (11), p.2082-2111
Main Authors: Giannetti, Flavia, Passarelli di Napoli, Antonia, Scheven, Christoph
Format: Article
Language:English
Subjects:
fractional Sobolev spaces
higher differentiability
partial regularity
stationary Stokes systems
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Summary:We consider weak solutions (u,π):Ω→Rn×R to stationary p‐Stokes systems of the type −diva(x,Eu)+∇π+[Du]u=f,divu=0in Ω⊂Rn, where the function a(x,ξ) satisfies p‐growth conditions in ξ and depends Hölder continuously on x. By Eu we denote the symmetric part of the gradient Du and we write [Du]u for the convective term. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient Eu and of the pressure π. As an application, we deduce dimension estimates for the singular set of the gradient Du, thereby improving known results on partial C1,α‐regularity for solutions to stationary p‐Stokes systems.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201800519