1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Maximum principle - don't understand why RHS is negative.

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider the linear inhomogeneous second order two point BVP;

    -a(x)u''(x) + b(x)u'(x) = f(x) for 0 < x < 1

    for some functions a, f, b where a(x) > 0 for all x

    1) If f(x) < 0 for x = [0,1], show that u(x) attains its maximum value at one of the two end points x = 0, 1. - I've done this, anyone who has done the maximum principle should be able to as well.

    2) Substitute v(x) = u(x) + [itex]\epsilon[/itex] [itex]e^{\lambda x}[/itex], show that if f(x) [itex]\leq[/itex] 0 then u(x) attains its maximum value at one of the end points x = 0,1


    3. The attempt at a solution

    Question two is what I'm stuck on. If we substitute v(x) in to the equation instead we get

    [PLAIN]http://img819.imageshack.us/img819/9646/unledmug.jpg [Broken]

    It says the RHS is strictly negative since a(x) > 0. How can we know this, when we don't know what sign b(x) may come out as for any x? It may turn out b(x) is positive and [itex]\lambda b[/itex] is greater than |[itex]\lambda^{2 } a[/itex]| and therefore what is contained within the brackets is positive, and may be bigger than the absolute value of f(x). Making the whole thing positive.

    Am I missing something?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 12, 2011 #2

    Deveno

    User Avatar
    Science Advisor

    for any x, b(x)λ is a linear function in λ.

    but a(x)λ2 is quadratic, so for sufficiently large λ, this term will dominate.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Maximum principle - don't understand why RHS is negative.
  1. Maximum principle (Replies: 4)

Loading...