Maximum principle for an elliptic system

Suppose $$\Omega$$ is a bounded smooth domain in $$R^N$$ and $$a,b$$ are bounded smooth vector fields in $$\Omega$$. Suppose we have $$-\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in } \Omega$$ $$-\Delta v(x) + b(x) \cdot \nabla u(x) = g(x) \ge 0 \quad \mbox{ in } \Omega$$ with $$u=v=0$$ on $$\partial \Omega$$. I am curious whether there is a maximum principle for $$u,v$$ in the sense that $$u,v \ge 0$$. I would prefer not to have any smallness or structural assumptions on $$a,b$$.

I assume these are well known results (just not to me). Any comments are greatly appreciated.

• If I understood correctly, I do not think there is any. Take $N = 1$, $\Omega = (0, \pi)$, $u(x) = -1 + \cos x$, $v(x) = -\sin x$, $a(x) = 1$, $b(x) = -1$ and $f(x) = g(x) = 0$. These satisfy the desired equations, but $u$ and $v$ are actually negative everywhere. Mar 2, 2019 at 7:58
• thank you very much for your answer...so with some smallness conditions on $a$ and $b$ i can prove (I think) but i wanted something better. If you add it as an answer i can accept it. Thanks again Mar 2, 2019 at 15:21
• Done. There was an error in my comment ($f(\pi) \ne 0$), but one can take $\Omega = (0, 2\pi)$ and everything works. Mar 2, 2019 at 21:06

1 Answer

Without any further assumptions, there is no such maximum principle. For example, in dimension $$N = 1$$, the functions $$u(x) = -1 + \cos x , \qquad v(x) = -\sin x$$ satisfy the system of elliptic equations given in the question with $$f(x) = g(x) = 0$$, $$a(x) = 1$$ and $$b(x) = -1$$, and both are equal to zero at the endpoints of $$\Omega = (0, 2\pi)$$. Nevertheless, neither $$u$$ nor $$v$$ is non-negative in $$\Omega$$.