We introduce integrands f : R nN → R of (s, µ, q)-type, which are, roughly speaking, of lower (upper) growth rate s ≥ 1 (q > 1) satisfying in additionwe prove partial C 1 -regularity of local minimizers u ∈ W 1 1,loc (Ω, R N ) by the way including integrands f being controlled by some N -function and also integrands of anisotropic power growth. Moreover, we extend the known results up to a certain limit and present examples which are not covered by the standard theory. (2000): 49 N 60, 49 N 99, 35 J 45
Mathematics Subject Classification
We investigate the smoothness properties of local solutions of the nonlinear Stokes problemwhere v: Ω → R n is the velocity field, π: Ω → R denotes the pressure function, and g: Ω → R n represents a system of volume forces, Ω denoting an open subset of R n . The tensor T is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1 < p ≤ q < ∞ such thatholds with suitable constants λ, Λ > 0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W 1 p,loc (Ω; R n ) are of class C 1,α on an open subset of Ω with full measure. If n = 2, then the set of interior singularities is empty.
Mathematics Subject Classification (2000). 76M30, 49N60, 35J50, 35Q30.
We extend the Liouville-type theorems of Gilbarg and Weinberger and of Koch, Nadirashvili, Seregin and Sverák valid for the stationary variant of the classical Navier-Stokes equations in 2D to the degenerate power law fluid model.
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