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PDE question: Eigenvalues

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Let λ_n denote the nth eigenvalue for the problem:

    -Δu = λu in A, u=0 on ∂A (*)

    which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions.

    (i) Show that (*) has no other eigenvalues

    (ii) Show that one can find a non-negative eigenfunction for the first eigenvalue of (*). Such an eigenfunction is also known as a ground state.
    Hint: If w is an eigenfunction that changes sign, then α=max(w,0) and β=min(w,0) are nonzero but of one sign; relate their Rayleigh quotients with that of their sum α+β=w


    Now let some real number 'a' be fixed and f(x) a given smooth function. Consider the problem:

    -Δu = au + f in A, u=0 on ∂A (**)

    (iii) Show (**) has no solution when a = λ_k is an eigenvalue of (*) and f is not orthogonal to the corresponding eigenfunction u_k.

    (iv) Let f(x) be a given smooth function and suppose a is not an eigenvalue of (*). Find the unique solution of the problem (**). Hint: the solution is a linear combination of the eigenfunctions u_k by completeness, so you need only determine the coefficients.

    2. Relevant equations



    3. The attempt at a solution
    Hi guys,

    Thanks for looking at my problem, I am a bit stuck on this particular one! We didn't do much about eigenvalues (with regard to pde's) in class and I haven't been able to find any books that give examples other than the basic theorems. Here's what I've done so far:

    (i) Use proof by contradiction. Let λ* be some other eigenvalue, ie λ_n < λ* < λ_(n+1)

    thus -Δu = λ*u

    for u a corresponding eigenfunction.

    The Rayleigh quotient is:

    R(u) = ( ∫|∇u|^2 ) / ( ∫∇u^2 )

    Suppose R(u) attains a minimum over the set:

    X_n = { u ∊ C^2(A): u=0 on ∂A, u ≢0 and u⊥u_1, ..,u_(n-1) }
    Then any function that minimizes R(u) over X_n is an eigenfunction u_n, with eigenvalue λ_n,

    where λ_n = min R(u), for u ∊ X_n.


    Now we want to show that λ* is not of this form i.e. minimizes R(u) but not over the set as outlined above.
    But if λ* is indeed an eigenvalue, and minimizes R(u), with corresponding eigenfunction u* say, then there will be some u ∊ {u_1, u_2, .... , u_n ... }, u_k, say, such that u_k ⊥ u*.
    But then u_k will be orthogonal to u_1, u_2, ..., u_(k-1), and then so will u*.

    But u* cannot be a member of this set.
    And this is a contradiction.

    Thus all λ are indeed of the required form.


    I don’t know if this is right (I think I may have left out something, it just seems a bit straightforward...) I would be really grateful if you could tell me if it is?

    (ii)
    By definition of the Rayleigh quotient,

    R(α) = ( ∫|∇ α |^2 ) / ( ∫∇ α ^2 )

    and

    R(β) = ( ∫|∇ β |^2 ) / ( ∫∇ β ^2 )

    and

    R(α +β) = ( ∫|∇ α |^2 + 2∫ ∇ α .∇ β + ∫|∇ β |^2 ) / ∫( α^2 + 2αβ + β^2)

    But then I’m stuck as to how to relate them, is it a case of minimizing R(α +β)?.



    (iii)
    If a= λ_k is an eigenvalue of (*), then:

    -Δu = au

    But -Δu = au+f

    So then au = au+f, unless f orthogonal (in which case it vanishes).
    So this gives us a contradiction otherwise.

    I know that seems really brief, but it is making sense in my head, I think I’m just not articulating it properly. Please could someone give me a point in the right direction?


    (iv)
    I actually have no idea how to go about solving this one. How do I go about setting up the equation and solving the coefficients?


    Thank you so much for looking at this and I would really appreciate any hints or pointers you could give me. Thanks again.
     
  2. jcsd
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