Let $U\subset \mathbb{R}^n$ be an bounded domain with smooth boundary. Assume $u$ is a smooth solution of $$ Lu=-\sum_{i,j=1}^na^{i,j}u_{x_ix_j}=f \ \text{in} \ U, \ \ u=0 \ \text{on} \ \partial U, $$ where $f$ is bounded. Fix $x^0 \in \partial U$. A $barrier$ at $x^0$ is a $C^2$ function $w$ such that $$ Lw\ge 1, \ \ w(x^0)=0, \ \ w|_{\partial U}\ge 0. $$ Show that if $w$ is a barrier at $x^0$, there exists a constant $C$ such that $$ |Du(x^0)|\le C|\frac{\partial w}{\partial \nu}(x^0)|. $$ Note that we assume $a^{i,j}$ are smooth and satisfy uniform ellipcity.
Actually, someone asked the problem on this website two years ago, but didn't get a clear answer.This is my homework and I would appreciate it if someone could help me. Thank you very much in advance.
