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.
Best Answer
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?