# Partial differential equation help needed (first order)

1. Sep 7, 2011

### dillingertaco

1. The problem statement, all variables and given/known data
Let u be a solution of $a(x,y)u_{x}+b(x,y)u_{y}=-u$ (I) of class $C^{1}$
in the closed unit disk $\Omega$ in the xy-plane. Let $a(x,y)x+b(x,y)y>0$ (II)
on the boundary of $\Omega$. Prove that u vanishes identically. (Hint: Show that on $\Omega$ max u$\leq 0$, min u $\geq 0$, using conditions for a maximum at a boundary point.

2. Relevant equations
$\frac{dx}{dt}=a(x,y)$
$\frac{dy}{dt}=b(x,y)$
$\frac{dz}{dt}=-z$
(the chapter was about solving first order pde's using autonomous systems of ODEs, but it doesn't seem to help here)

3. The attempt at a solution
First the closed ball is compact, so a C^1 function attains a max and a min. If this max/min occurs in the inside of Omega, then each partial derivative equals 0, and by (I) u is identically 0. If the max/min occur on the boundary, then the partial derivatives do not necessarily equal zero, and that is the case i'm struggling with.

I know I want to use (II) to change the problem but there's no way I can think of to guarantee $u_{x}=x\ u_{y}=y$ (to just plug in and say -u is greater than 0, so I tried this:
By adding (II) since it is greater than zero we know:
$-u = a(x,y)u_{x}+b(x,y)u_{y} < a(x,y)(u_{x}+x)+b(x,y)(u_{y}+y)$
and
$-u > a(x,y)(u_{x}-x)+b(x,y)(u_{y}-y)$

I didn't quite see anything here, so i went another direction using the hint, the condition at a boundary is that for all u, $u_{max}\geq u$. Not sure how to use this. I also was thinking it had to do that a maximum is in the direction of the partial derivatives, but i can't see where to go with that. We also know on the boundary $x^{2}+y^{2}=1$ but again no idea what to use it for, the most i can see is starting with:

$-u=a(x,y)u_{x}(1)+b(x,y)u_{y}(1)=a(x,y)u_{x}(x^{2}+y^{2})+b(x,y)u_{y}(x^{2}+y^{2})=x^{2}(-u)+y^{2}(-u)$

I also tried assuming u was nonzero somewhere for a contradiction.. drawing a blank.

anyone got any ideas??

Last edited: Sep 7, 2011