Abstract
We study regularity results for nonlinear parabolic systems of p-Laplacian type with inhomogeneous boundary and initial data, with p is an element of(2nn+2,infinity) We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderon-Zygmund theory for such systems.
This extends the global result of Bogelein (Calc Var Partial Differ Equ 51(3-4):555-596, 2014) to very rough domains and more general boundary values. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results.
Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, p is an element of(2nn+2,2]any scaling deficit can be omitted.