We have two real-valued functions $f, g: \mathbb{R}^2\rightarrow\mathbb{R}$. Say a PDE $$ f(x,y)\partial_x u(x,y) + g(x,y)\partial_yu(x,y) = -u(x,y) $$ defined in $B_1(0)$ has a continuous solution $u(x,y)$. If on the boundary, we prescribe some data which satisfies $f(x,y)x + g(x,y)y > 0$, prove that $\max_{B_1(0)} u \leq 0$ and $\min_{B_1(0)} u \geq 0$, hence $u$ is constantly zero everywhere in $B_1(0)$.
I tried to use characteristics to proceed, but without specific knowledge of the functions and boundary data, it is leading me nowhere. I just do not see how can I incorporate max and min to the game here.
Any helps with this problem?
I think what the correction notation shall be sup and inf?
Try to figure out what the boundary data tells you about the direction of the characteristics.
Then the $-u$ term shall tell you how the function behaves as it moves along characteristics.
Last, you can search for contradictions if sup $> 0$ and if inf $< 0$.
You might find it helpful to separate cases where sup/inf is attained inside and on the boundary.