Timeline for Solve |\nabla u|^2=1

6 events

when toggle format	what		by	license	comment
Dec 13, 2011 at 10:18	vote	accept	Mahdiyar
Dec 13, 2011 at 10:10	comment	added	Mahdiyar		Thanks Robert. I guess you meant $F'=f$ not $=1/f$. I would've picked this as my answer if it were not a comment.
Dec 13, 2011 at 6:16	answer	added	Denis Serre		timeline score: 2
Dec 12, 2011 at 20:10	answer	added	Rebs		timeline score: 2
Dec 12, 2011 at 20:01	comment	added	Robert Bryant		@Mahdiyar: Look up the literature on the eikonal equation, which this is. There, you will see that the `general solution' is to let $u(x,y)$ be a constant plus the (signed) distance of $(x,y)$ to some set (usually a point or a curve) in the plane. The only global smooth solutions are linear functions (as you gave), and the only global solution with only one singular point is the distance from a point (plus constants), as you saw. For the more general case, as long as $f$ is strictly positive, you let $u = F(v)$ where $v$ satisfies the original equation ($f\equiv1$) and $F'(v) = 1/f(F(v))$.
Dec 12, 2011 at 19:45	history	asked	Mahdiyar	CC BY-SA 3.0