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Can please somebody tell me, how solve this problem ?

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 $(v^2w_{x_i})_{x_j}$.)

Thanks so much in advance.

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    $\begingroup$ You're presenting a (graduate level?) qualifying exam, but you can't explain anything about what you've tried and what your thoughts are? $\endgroup$ – Erick Wong Jan 4 '14 at 0:36
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First, assume that you found the operator $M$ in the hint. If $w$ has a local maximum, all first derivatives are zero and all second derivatives are negative, so $Mw$ is positive, a contradiction. Thus, all maxima are on the boundary, so $w\leq 0$. (Edit: I implicitly assumed that the mixed partial terms were 0, but the result still holds because the second derivative terms can be diagonalized)

But $v$ is positive, so $u=vw$ must also be nonnegative.

This assumes that you completed the hint. Would you like help with that part?

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  • $\begingroup$ Thank you for your answer. Actually, the hint is the difficult part for me, could you please help me with that? $\endgroup$ – FASCH Jan 4 '14 at 1:58
  • $\begingroup$ @FASCH In order for someone to help you with that, you might want to explain what you're having trouble with. $\endgroup$ – Erick Wong Jan 4 '14 at 5:26
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    $\begingroup$ Dear Brian Rushton, thanks again for you answer. I already got the operator $M$ for myself. $\endgroup$ – FASCH Jan 5 '14 at 1:10
  • $\begingroup$ @BrianRushton If you can elaborate the hint, please let me know. My question (math.stackexchange.com/questions/1148857/weak-maximum-principle) concerns precisely this part only. I do believe I can complete the exercise on my own after figuring out the operator $M$. $\endgroup$ – Cookie Feb 16 '15 at 22:44
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$Mw=-\sum_{ij=1}^{n}a^{ij}w_{ij}+\sum_{i=1}^{n}(b_i -\frac{2}{w}\sum_{k=1}^{n}a^{ik}v_k)w_i$

$Mw=\frac{Lu}{v}-\frac{uLv}{v^2}$

Use this we can solve the problem, I find in Han Qing's book.

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    $\begingroup$ It may be $Mw=-\sum_{ij=1}^{n}a^{ij}w_{ij}+\sum_{i=1}^{n}(b_i -\frac{2}{\color{red}{v}}\sum_{k=1}^{n}a^{ik}v_k)w_i$ $\endgroup$ – Hans May 16 '15 at 14:00

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