A regularity theory for a more general class of quasilinear parabolic partial differential equations and variational inequalities

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that solutions to degenerate parabolic variational inequalities are locally Hölder continuous. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.