1. Sep 15, 2009

### Kizaru

I can't really find much clarification on Hadamard's definition of a well-posed problem.
My confusion comes from knowing exactly what is meant by the second and third properties:
2) The solution is unique
3) The solution depends continuously on the data, in some reasonable topology. http://en.wikipedia.org/wiki/Well-posed_problem" [Broken]

For example, a solution would exist for a PDE y * uxx = x * uyy with u(0,y) = 2 and u(x,0) = 2 that is simply a constant. But is the solution, u=2, considered unique? Does the solution depend on the data?

Does the solution for u(x,y) fail property 3 because it no longer satisfies the PDE if the initial conditions are changed?

Edit: Maybe this belongs in the Differential Equations forum. Sorry. Can this be moved? Thanks.

Last edited by a moderator: May 4, 2017
2. Sep 15, 2009

### HallsofIvy

Staff Emeritus

Is there any other function of x and y that will satisfy those conditions? If not, then it is unique.

Well suppose the problem were exactly the same differential equation but u(0,y)= 3 and u(x,0)= 3. Then u(x,y)= 3 is a solution. Since changing the "data" changes the solution, yes, the solution depends on the data.

No, property 3 does not say that the solution must be independent of the initial conditions, it say it must depend on them continuously. suppose the intial conditions were $u(x, 0)= 2+\delta$, $u(0, y)= 2+ \delta$. Then [/itex]u(x,y)= 2+ \delta[/ itex] is obviously a solution. And, as $\delta$ goes to 0, that solution goes to u(x,y)= 2. In that situation, the solution depends continuously on the initial values.

Okay, I will move it.

Last edited by a moderator: May 4, 2017
3. Sep 15, 2009