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.
Highlights
Consider the Euclidean space Rn with integer n ≥ 1
In this paper we investigate local gradient estimates for weak solutions to equations of divergence form div[a(x, u, ∇u)] + b(x, u, ∇u) = 0 in B3, where the vector field a and the function b satisfy certain ellipticity and growth conditions
Let K ⊂ R be an interval, and let a = (a1, . . . , an) : B3 × K × Rn → Rn and b : B3×K×Rn → R be Caratheodory maps such that a is differentiable on B3×K×(Rn \{0})
Summary
Consider the Euclidean space Rn with integer n ≥ 1. Our main motivation for deriving the local gradient estimates in Theorems 1.2 and 1.3 is to be able to establish W 1,q-estimates (for q > p) for weak solutions to a large class of equations of the form div A(x, u, ∇u) + B(x, u, ∇u) = div F, where the vector field A is allowed to be discontinuous in x, Lipschitz continuous in u and its growth in the gradient variable is like the p-Laplace operator with 1 < p < ∞ This is achieved in our forthcoming work [6] by using Caffarelli-Peral perturbation technique [2], and the quantified estimate (1.4) for (1.1) plays an essential role in performing that process. Some lower order terms arising from the x, z dependence are treated carefully and differently (see (2.2) below) compared to the known work in order to obtain the desired homogeneous estimate
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