Local Gradient Estimates for Degenerate Elliptic Equations

This paper is focused on the local interior W-1,W-infinity-regularity for weak solutions of degenerate elliptic equations of the form div[a(x, u, del u)] + b(x, u, del u) = 0, which include those of p-Laplacian type. We derive an explicit estimate of the local L-infinity-norm for the solution's gradient in terms of its local L-p-norm. Specifically, we prove parallel to del u parallel to(p)(L infinity(BR/2(x0)) <= C/vertical bar B-R(x(0))vertical bar integral(BR(x0)) vertical bar del u(x)vertical bar(p) dx. This estimate paves the way for our work [9] in establishing W-1,W-q-estimates (for q > p) for weak solutions to a much larger class of quasilinear elliptic equations.