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I am stuck with the following problem:

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

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what is $+_{yy}$? –  Paul Dec 16 '12 at 11:53
    
$u_{yy}=\frac{\delta ^{2}u}{\delta y^2}$ –  user52976 Dec 16 '12 at 12:01
    
Originally you wrote $u_{xx}+_{yy}+\lambda u=0$. Then Jon corrected your typo. –  Paul Dec 16 '12 at 12:27
    
You are looking for the eigenvalues/eingefunctions of $\Delta$ in a square. You should read the notes of this course: math.ucdavis.edu/~saito/courses/LapEig –  Siminore Dec 16 '12 at 13:37
    
Thanks a lot Jon and Paul. –  user52976 Dec 16 '12 at 14:10

2 Answers 2

Consider the basis $$ B=\{\phi_{00},\ \theta_{0n},\ \xi_{n0},\ \phi_{mn},\ \psi_{mn},\ \theta_{mn},\xi_{mn}\}_{(m,n) \in \mathbb{N}^2} $$ of $L^2([0,1]^2)$ given by \begin{eqnarray} \phi_{00}(x,y)&=&1,\ \theta_{0n}(x,y)=\sin(2\pi ny),\ \xi_{n0}(x)=\sin(2\pi nx),\\ \phi_{mn}(x,y)&=&\cos(2\pi mx)\cos(2\pi ny),\ \psi_{mn}(x,y)=\sin(2\pi mx)\sin(2\pi ny),\\ \theta_{mn}(x,y)&=&\cos(2\pi mx)\sin(2\pi ny),\ \xi_{mn}(x,y)=\sin(2\pi mx)\cos(2\pi ny). \end{eqnarray} Then for every $m,n \ge 0$ we have $$ -\Delta\phi_{mn}=\lambda_{mn}\phi_{mn},\ -\Delta\psi_{mn}=\lambda_{mn}\psi_{mn},\ -\Delta\theta_{mn}=\lambda_{mn}\theta_{mn},\ -\Delta\xi_{mn}=\lambda_{mn}\xi_{mn}\ \forall\ m,n \ge 0, $$ where $$ \lambda_{mn}=4\pi^2(m^2+n^2) \quad \forall\ m, n\ge 0. $$ This shows that the number $\lambda_{mn}$ is an eigenvalue of $-\Delta$ with corresponding eigenfunctions $\phi_{mn},\psi_{mn},\theta_{mn},\xi_{mn}$.

Because of the boundary conditions \begin{eqnarray} u(x,0)=u(x,1)&=&0 \quad \forall x \in [0,1],\\ u(0,y)=u(1,y)&=&0 \quad \forall y \in [0,1], \end{eqnarray} it is clear that a pair $(u,\lambda)$ satisfies the given PDE precisely when it belongs to $\{(c\psi_{mn},\lambda_{mn}): \ m,n \ge 1, \ c \in \mathbb{R}\setminus\{0\}\}$.

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Hint: See the Dirichlet eigenvalues.

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Thanks a lot sir.It has been useful. –  user52976 Dec 16 '12 at 17:34
    
@user52976: You are welcome. –  Mhenni Benghorbal Dec 16 '12 at 17:39

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