Local Gradient Estimates for Degenerate Elliptic Equations

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