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On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided ${u|}_{\partial B}=0$ and ${lim}_{\left|x\right|\to \infty }{\left|x\right|}^{1/3}\left|u\left(x\right)\right|=0$ uniformly. For slow flows the latter condition can be replaced by ${lim}_{\left|x\right|\to \infty }\left|u\left(x\right)\right|=0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.

We consider local minimizers $u:{ℝ}^2\supset \Omega \to {ℝ}^N$ of variational integrals like ${\int }_{\Omega }\left[\right(1+|{\partial }_1{u|}^2{)}^{p/2}+(1+|{\partial }_2{u|}^2{)}^{q/2}]dx$ or its degenerate variant ${\int }_{\Omega }\left[\right|{\partial }_1{u|}^p+|{\partial }_{2}u{|}^q]dx$ with exponents $2\le p<q<\infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. (2003), 177–186. We prove interior $C^{1,\alpha }$- respectively $C^1$-regularity of $u$ under the condition that $q<2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. (2006), 349–362.

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p,q)$-growth with exponents $p\le q\&lt;\infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1,\mu }$. Note that to our knowledge the only $C^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.