A Global Lp-Gradient Estimate on Weak Solutions of Nonlinear Parabolic Systems under Mixed Boundary Conditions

Abstract

In this paper, we prove the integrability of the spatial gradient Du to an exponent p>2 near the boundary, u being a weak solution of a nonlinear parabolic system under mixed boundary conditions. Our method of proof relies on an adaption of a technique by Gehring-Giaquinta-Modica (higher integrability by reverse Hölder inequality) to cubes which possibly intersect a hyperplane. Combined with the interior estimate in (Naumann, J., Wolff, M.: Interior integral estimates on weak solutions of nonlinear parabolic systems. - Preprint no. 94-12; Fachber. Math., Humboldt-Univ.) we obtain a global Lp-gradient estimate.