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I have trouble solving the following Evans' PDE problem. I would appreciate it if someone could help me solving it. Thank you very much in advance.

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.

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    Fixed the typo in the definition.2012-12-14

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Give two functions $v_{1}= u + \Vert f \Vert_{\infty} w$, $v_{2} = -u + \Vert f \Vert_{\infty} w$. Now, calculate $Lv_1, Lv_2$, and then use maximum principle to obtain the answer .....

Good Luck

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Note that $\nabla u(x_0)$ and $\nabla w(x_0)$ are parallel to each other and to $\nu(x_0)$.

Suppose you have somehow found two $C^2$ functions $w_1$ and $w_2$ with the properties $w_1(x_0) = 0 = w_2(x_0)$ , $w_1 \ge 0 \ge w_2$ on $\partial \Omega$, and $Lw_1 \ge Lu \ge Lw_2$ in $\Omega$.

What can you say about $w_1, w_2, u$? What can you say about their gradients at $x_0$? And can you perhaps construct such functions from what you have?

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    I now see your argument. Weakly minimum principle implies $x_0$ is the minimum of $w$ on $\overline{U}$.2012-12-15