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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.

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1 Answers1

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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 $\{x \in \overline U ; u > 0\}$.

Aaron Maroja
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