We say that the uniformly elliptic operator
$$Lu\ =\ -\sum_{i,j=1}^na^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$
satisfies the weak maximum principle if for all $u\in C^2(U)\cap C(\bar{U})$
$$\left\{\begin{array}{rl}
Lu \leq 0 & \mbox{in } U\\
u \leq 0 & \mbox{on } \partial U
\end{array}\right.$$
implies that $u\leq 0$ in $U$.Suppose that there exists a function $v\in C^2(U)\cap C(\bar{U})$ such that $Lv \geq 0$ in $U$ and $v > 0$ on $\bar{U}$. Show the $L$ satisfies the weak maximum principle.
(Hint: Find an elliptic operator $M$ with no zeroth-order term such that $w := u/v$ satisfies $Mw \leq 0$ in the region $\{u > 0\}$. To do this, first compute $(v^2w_{x_i})_{x_j}$.)
This is from PDE Evans, 2nd edition: Chapter 6, Exercise 12.
A question has been asked already about this problem, but my question is not considered a duplicate of it. That other question asks to solve the problem altogether; my question about the problem is merely finding the elliptic operator $M$, which is not explained thoroughly in the other question.
How can I construct the elliptic operator $M$? I am following the hint given by computing $$(v^2w_{x_i})_{x_j}=2vv_{x_j}w_{x_i}+v^2w_{x_i x_j}.$$
Now, I do not know what to do here honestly, but I thought about saying
$$Mw = -\sum_{i,j=1}^n v^2 w_{x_i x_j} – \sum_{i=1}^n 2vv_{x_j}w_{x_i}-cu,$$
so that $Mw \le 0$, when considering that "$w:=u/v$ satisfies $Mw\le 0$ in the region $\{u > 0\}$".
As soon as I receive confirmation that my $M$ is fine, then from this point on I can complete the exercise on my own.
Best Answer
Let $u \in C^2 (U) \cap C(\overline U)$ be given that solves the PDE. We have that $\overline U$ is compact (U is bounded) and $v \in C(\overline U)$, $v \geq c > 0$. Let $w = \frac{u}{v}$, then $w \in C^2 (U) \cap C(\overline U)$.
Through brute force we differentiate $w$ to get
$$-a^{ij} w_{x_i} = - a^{ij} \frac{u_{x_i}}{v} + a^{ij}\frac{uv_{x_i}}{v^2}$$
$$\begin{align}-a^{ij} w_{x_ix_j} &= \frac{-a^{ij} u_{x_ix_j} v + a^{ij} uv_{x_ix_j} }{v^2} + a^{ij} \frac{2}{v}v_{x_j} \color{red}{w_{x_i}}\end{align}$$
Therefore
$$-\sum_{i,j=1}^n a^{ij} w_{x_ix_j} = \frac{Lu}{v} - \frac{uLv}{v^2} - \sum_i b^iw_{x_i} + \frac{2}{v}\sum_{i,j} a^{ij} v_{x_j}w_{x_i} $$
Now the operator we are looking for is defined as
$$\bbox[3px,border:2px solid red] {Mw = -\sum_{i,j=1}^n a^{ij} w_{x_ix_j} + \sum_i\left[b^i - \sum_{j=1}^n \frac{2}{v}a^{ij}v_{x_j}\right]w_{x_i} }$$
This answers your question.
Remark: This definition is taken so we obtain
$$Mw = \frac{Lu}{v} - \frac{uLv}{v^2} \leq 0$$ on $\{x \in \overline U ; u > 0\}$.