Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hadamard and well-posed problems.

  1. Sep 15, 2009 #1
    Hadamard and "well-posed" problems.

    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. jcsd
  3. Sep 15, 2009 #2


    User Avatar
    Science Advisor

    Re: Hadamard and "well-posed" problems.

    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 [itex]u(x, 0)= 2+\delta[/itex], [itex]u(0, y)= 2+ \delta[/itex]. Then [/itex]u(x,y)= 2+ \delta[/ itex] is obviously a solution. And, as [itex]\delta[/itex] 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
  4. Sep 15, 2009 #3
    Re: Hadamard and "well-posed" problems.

    Thank you so much! It's much clearer than my professor's explanation :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook