GRADIENT ESTIMATES FOR ENTROPY SOLUTIONS TO ELLIPTIC EQUATIONS WITH DEGENERATE COERCIVITY

We derive some a priori estimates and then prove the existence and regularity of distributional solutions to degenerate elliptic equations of the form ( -div d(x, u(x), backward difference u(x)) = f (x), x is an element of center dot, u(x) =0, x is an element of partial differential center dot, where the Caratheodory function d : center dot x IIB x IIBn -> IIBn satisfies |xi|p d(x,s, xi)center dot xi >= alpha (1 + |s|)theta , |d(x,s, xi)| <=beta|xi|p-1 for 1 < p < n, 0 <= theta < p -1 and 0 < alpha <= beta < infinity, with f a Marcinkiewicz function. We obtain a gradient estimate for entropy solutions, and show that the result is optimal by a counterexample. We also consider degenerate elliptic equations with a lower order term, ( -div d(x, u(x), backward difference u(x))+ |u|r-1u = f (x), x is an element of center dot, u(x) = 0, x is an element of partial differential center dot. We show that the presence of the lower order term has regularizing effects on our result.