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I am stuck with the following problem:

The PDE
$u_{xx}+u_{yy}+\lambda u=0, 0
$u(x,0)=u(x,1)=0; 0\leq x \leq 1$
$u(0,y)=u(1,y)=0; 0\leq y \leq 1$
has

(a)a unique solution u for any $\lambda \in \mathbb R ,$
(b)infinitely many solutions for some $\lambda \in \mathbb R ,$
(c)a solution for countably many values of $\lambda \in \mathbb R ,$
(d)infinitely many solutions for all $\lambda \in \mathbb R .$

I do not know how to progress with it.Could someone point me in the right direction( e.g. a certain theorem or property i have to use?) Thanks in advance for your time.

  • 0
    what is $+_{yy}$?2012-12-16
  • 0
    $u_{yy}=\frac{\delta ^{2}u}{\delta y^2}$2012-12-16
  • 0
    Originally you wrote $u_{xx}+_{yy}+\lambda u=0$. Then Jon corrected your typo.2012-12-16
  • 0
    You are looking for the eigenvalues/eingefunctions of $\Delta$ in a square. You should read the notes of this course: http://www.math.ucdavis.edu/~saito/courses/LapEig/2012-12-16
  • 0
    Thanks a lot Jon and Paul.2012-12-16

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