Geodesic
Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 493-508

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We discuss regularity results concerning local minimizers $u: \mathbb R^n\supset \Omega\rightarrow\mathbb R^n$ of variational integrals like \begin{align*} \int_{\Omega}\{F(\cdot ,\varepsilon (w))-f\cdot w\}\,dx \end{align*} defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset $\Omega_{0}$ of $\Omega$ with full measure if $q p\,\frac{n+2}{n}$ (for $n=2$ we have $\Omega_{0}=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear Stokes type system.
Classification : 35B65, 35J50, 35Q35, 49N60, 76D07, 76M30
Keywords: Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer 
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     author = {Breit, Dominic},
     title = {Smoothness properties of solutions to the nonlinear {Stokes} problem with nonautonomous potentials},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {493--508},
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Breit, Dominic. Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 493-508. http://geodesic.mathdoc.fr/item/CMUC_2013__54_4_a3/