Local Gradient Estimates for Degenerate Elliptic Equations

Abstract

Abstract This paper is focused on the local interior W 1 , ∞ ${W^{1,\infty}}$ -regularity for weak solutions of degenerate elliptic equations of the form div ⁡ [ 𝐚 ⁢ ( x , u , ∇ ⁡ u ) ] + b ⁢ ( x , u , ∇ ⁡ u ) = 0 ${\operatorname{div}[\mathbf{a}(x,u,\nabla u)]+b(x,u,\nabla u)=0}$ , which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞ ${L^{\infty}}$ -norm for the solution’s gradient in terms of its local L p ${L^{p}}$ -norm. Specifically, we prove ∥ ∇ ⁡ u ∥ L ∞ ⁢ ( B R / 2 ⁢ ( x 0 ) ) p ≤ C | B R ⁢ ( x 0 ) | ⁢ ∫ B R ⁢ ( x 0 ) | ∇ ⁡ u ⁢ ( x ) | p ⁢ 𝑑 x . $\lVert\nabla u\rVert_{L^{\infty}(B_{R/2}(x_{0}))}^{p}\leq\frac{C}{\lvert B_{R}% (x_{0})\rvert}\int_{B_{R}(x_{0})}\lvert\nabla u(x)\rvert^{p}dx.$ This estimate paves the way for our work [9] in establishing W 1 , q ${W^{1,q}}$ -estimates (for q > p ${q>p}$ ) for weak solutions to a much larger class of quasilinear elliptic equations.