Global Gradient Estimates for Nonlinear Elliptic Equations with Vanishing Neumann Data in a Convex Domain

In this paper, we obtain the following global L ^q estimates

$$\left|\mathbf{f}\right|^{p } \in L^{q}({\Omega}) \Rightarrow \left|\nabla u\right|^{p } \in L^{q}({\Omega}) \quad \text{for any} ~~q\ge 1 $$

in a convex domain Ω of weak solutions for nonlinear elliptic equations of p-Laplacian type with vanishing Neumann data

$$\begin{array}{@{}rcl@{}} \text{div} \left( \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \right) & =& \text{div} \left( | \mathbf{f}|^{p-2} \mathbf{f} \right) \quad\text{in} ~~{\Omega},\\ \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \cdot \mathbf{\nu} &=& | \mathbf{f}|^{p -2} \mathbf{f}\cdot \mathbf{\nu} \quad \quad \text{on}~~ \partial{\Omega}, \end{array} $$

where ν is the outwardpointing unit normal to ?Ω. Our argument is based on the works of Banerjee and Lewis (Nonlinear Anal 100:78–85, 2014), Kinnunen and Zhou (Comm Partial Differential Equations 24(11&12):2043–2068, 1999, Differential and Integral Equations 14(4):475–492, 2001), and Byun, Wang, and Zhou (Comm Pure Appl Math 57(10):1283–1310, 2004, J Funct Anal 20(3):617–637, 2007). In the proof of the above result, we only focus on the boundary case while the interior case can be obtained as a corollary.