Variants of the Stokes Problem: the Case of Anisotropic Potentials

We investigate the smoothness properties of local solutions ofthe nonlinear Stokes problem$begin{eqnarray*}-diverg {T(eps(v))} + nabla pi &=& g msp mbox{on $Omega$,}diverg v&equiv & 0 msp mbox{on $Omega$,}end{eqnarray*}$where v: ⁿ is the velocity field, $pi$: $ denotes the pressurefunction, and g: ⁿ represents a system of volume forces, denotingan open subset of ⁿ . The tensor T is assumed to be the gradient of some potential facting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents1 < p q < infty such thatlambda (1+|eps|^{2})^{frac{p-2}{2}} |sigma|^{2} leq D^{2}f(eps)(sigma ,sigma)leq Lambda (1+|eps|^{2})^{frac{q-2}{2}} |sigma|^{2}holds with suitable constants , > 0, i.e. the potential f is of anisotropicpower growth. Under natural assumptions on p and q we prove that velocity fields fromthe space W ¹ _{p, loc}(; ⁿ )are of class C ^1, on an open subsetof with full measure. If n = 2, then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday