A regularity result for quasilinear parabolic systems

Abstract.

We consider bounded, weak solutions of certain quasilinear parabolic systems of second order. If the solution fulfills a suitable smallness condition, we show that it is Hölder continuous and satisfies an a priori estimate. This is a well known result of Giaquinta and Struwe [3]. Their argument employs the use of Green’s functions, which is completely avoided in our proof. Instead, our crucial tool is a weak Harnack inequality for supersolutions due to Trudinger [7] in connection with a technique developed by L.Caffarelli [1].