I would be very glad if somebody could give me some advice on how to solve this problem:
"Let $f\in C^1(\mathbb{R}^2,\mathbb{R})$ with $\partial_1f(x,t)=\partial_2 f(x,t)$ $\forall (x,t)\in\mathbb{R}^2$ and $f(x,0) > 0$ $\forall x$. Show that $f(x,t)>0$ $\forall (x,t)\in\mathbb{R}^2$."
I'm struggling to even find an approach to this problem. Normally, in the "2d" case, one could just define a new function which contains $f$ in some way and then apply the Mean value theorem or Rolle's theorem or something like that. I'm not quite sure whether it works in this case, I tried several functions but they didn't seem to work, so I guess those times are over. Does someone have a tip for me what I could try out? Thanks in advance!
What you have is a transport equation.
Observe that $\partial_1 f-\partial_2 f=\nabla f\cdot (1,-1)$ is just the directional derivative of $f$ in the direction $(1,-1)$. Your PDE means that your function is a constant as you move in this direction. But moving in this direction starting at some point on the axis $t=0$ takes you to any point in $\mathbb{R}^2$, so your function is positive everywhere.