TITLE: The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
QUESTION [21 upvotes]: I am interested in solving the following biharmonic eigenvalue problem.   
$$\begin{array}{cccc}
  & \Delta ^2  \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a &  - b \le y \le b \\
  & x = \phantom{-}a & \Psi = 0 &   \dfrac{\partial \Psi }{\partial x} = 0 \\
  & x =  - a & \Psi = 0 &  \dfrac{\partial  \Psi }{\partial x} = 0 \\
  & y = \phantom{-}b & \Psi = 0 &  \dfrac{\partial  \Psi }{ \partial y} = 0 \\
 & y =  - b & \Psi = 0 &  \dfrac{\partial  \Psi }{ \partial y} = 0 
\end{array}
$$
where
$$ \Delta ^2 \Psi  = \frac{\partial ^4 \Psi }{\partial x^4} + 2 \frac{\partial^4 \Psi }{\partial x^2 \partial y^2} + \frac{\partial ^4 \Psi }{\partial y^4}$$
$$\Psi  \in {{\bf{C}}^{\infty}}\left( {[ - a,a] \times [ - b,b]} \right)$$
To describe the problem in words, we are looking for the eigenfunctions of the biharmonic operator over a rectangular domain where all its derivatives are continuous. The boundary conditions are of Dirichlet type, i.e., the function and it's normal derivative are prescribed over the boundary of the rectangular domain.

Facts and Motivations 
1) This problem occurs in many physical areas. One of the most famous ones is the vibration of a rectangular isotropic elastic clamp plate. 
2) It is believed between the engineers that the problem doesn't have a closed form solution. It may be asked that even the problem has a solution or not. Numerical evidence shows that such a solution may exists. However, I am looking for some strong theoretical basis to prove the existence of the solution so I planned to ask this question in a society of mathematicians. 

Questions
1) Is there any non-zero solution for this problem? In other words, I am asking an existence or non-existence theorem for this problem.
This question is completely answered by TKS. According to TKS, it is an old result firstly proved by K. Friedrichs. Maybe the reason that many people are unaware of this is that the paper by K. Friedrichs is written in German entitled as 
Die Randwert- und Eigenwertprobleme aus der Theorie der elastischen Platten. (Anwendung der direkten Methoden der Variationsrechnung)
The translation in English is 
The boundary value and eigenvalue problems in the theory of elastic plates. (Application of direct methods of variational calculus)
Another short answer to this question is given by Jean Duchon on Math Over Flow.
2) Assuming the existence, how can one compute these eigenvalues and eigenfunctions? Is there a closed form solution for this purpose?
This question remained unanswered!

REPLY [4 votes]: The work done earlier as below is similar to what you are attempting (except may be the coupling of different orders). Features are non-orthogonal eigen-functions $F_k(y)$satisfying boundary conditions of Biharmonic Equation, supplying characteristic equation yielding eigen values $\rho_k$ developed in $k$ Fadle- Popkovich expansion procedure using auxiliary clamped beam functions $Y_m$ to successfully orthogonalize and evaluate all coefficients including $Y_m$ numerically. 
$$ \sum A_k a_{km} = C_m ;\qquad \sum A_k a_{km} \rho_k = D_m ; $$
The satisfaction of zero deflection at corners is found accurate to the order $10^{-6}$. 
Coefficient $ a_{km}$ evaluated is useful for your work too hopefully. Full details are in the following paper Elsevier published  original research.
Orthotropic plate Eigen function Expansions