A Degenerate Neumann Problem for Quasilinear Elliptic Equations

Abstract

The degenerate Neumann problem \[ \begin{cases} \ \displaystyle \sum_{i,j=1}^{n}a^{ij}(x)\frac{\partial^{2}u}{\partial x_i\partial x_j}=f(x,u,Du) & \text{in}\ \Omega ,\\ \ a(x)\dfrac{\partial u}{\partial v}+b(x)u=\varphi(x) & \text{on}\ \Gamma \end{cases} \] is studied in the case where $a(x)$ and $b(x)$ are non-negative functions on $\Gamma$ such that $a(x)+b(x)>0$ on $\Gamma$. A classical existence and uniqueness theorem in the Hölder space $C^{2+\alpha}(\bar{\Omega})$ is proved under suitable regularity and structure conditions on the data.