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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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 
@article{CMUC_2013_54_4_a3,
     author = {Breit, Dominic},
     title = {Smoothness properties of solutions to the nonlinear {Stokes} problem with nonautonomous potentials},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {493--508},
     year = {2013},
     volume = {54},
     number = {4},
     mrnumber = {3125072},
     zbl = {06373980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2013_54_4_a3/}
}
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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/