Jens Habermann

Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients

Abstract. We establish partial regularity for solutions to systems modeling electro-rheological fluids in the stationary case. As a model case our result covers the low order regularity of systems of the type where denotes the symmetric part of the gradient , denotes the pressure, the not necessarily continuous coefficient is a bounded non-negative -function and the variable exponent function fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity of the exponent function there holds To be more precise, we prove Hölder continuity of the solution outside of a negligible set. Moreover, we show that and the pressure belong to certain Morrey spaces on the regular set of , i.e., the set where is Hölder continuous. Note that under such weak assumptions partial Hölder continuity for the gradient cannot be expected. Our result is even new if the coefficient is continuous.

Global higher integrability for non-quadratic parabolic quasi-minimizers on metric measure spaces

Abstract We prove up-to-the-boundary higher integrability estimates for parabolic quasi-minimizers on a domain Ω T \Omega_{T} = = Ω × \times (0,T), where Ω denotes an open domain in a doubling metric measure space which supports a Poincaré inequality. The higher integrability for upper gradients is shown globally and under optimal conditions on the boundary ∂ \partial Ω of the domain as well as on the boundary data itself. This is a starting point for a further discussion on parabolic quasi-minima on metric measure spaces, such as for example regularity or stability issues.