Harnack's inequality and H'older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

摘要

We continue to study regularity results for weak solutions of the large classof second order degenerate quasilinear equations of the form \begin{eqnarray}\text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for}x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper givinglocal boundedness of weak solutions. Here we derive a version of Harnack'sinequality as well as local H\"older continuity for weak solutions. Thepossible degeneracy of an equation in the class is expressed in terms of anonnegative definite quadratic form associated with its principal part. Nosmoothness is required of either the quadratic form or the coefficients of theequation. Our results extend ones obtained by J. Serrin and N. Trudinger forquasilinear equations, as well as ones for subelliptic linear equationsobtained by Sawyer and Wheeden in their 2006 AMS memoir article.