Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$.
If I know that \begin{align} &\bullet\quad -\Delta y_i=1 \text{ for } i=1,2 \text{ on } B\\ &\bullet\quad 0\leq y_1(x)\leq y_2(x) \text{ on } \bar{B}\\ \end{align} Does it mean there is a $x\in \bar{B}$ such that $|\nabla y_1(x)|\leq |\nabla y_2(x)|$?
The usual gradient estimates from Gilbard-Trudinger do not seem to help. Thanks in advance.
I think you should be able to do this.
The idea is that you let $y_2$ be as flat as possible, with large boundary values (so $y_2 = \frac{x^2}{2n} + M$ for some large $M > 0$).
Then you let $y_1 = \frac{x^2}{2n} + B \cdot x + C$ for a very large value of $B$ and corresponding $C$, so that the term in the gradient from $B$ overwhelms the one from the quadratic term. This is not so large $y_1$ is ever allowed to be greater than $M$ - since you can make $M$ as large as you like, this works.