I will be so thankful if you provide some links! :)

Also, It's a biharmonic eigenvalue problem not a Laplacian one. :)

Can you solve your problem with a finite superposition of plane waves, including possibly evanescent waves? If not, I would guess it cannot be solved, or at the very least, it should be extremely difficult.

@ everybody: Please see the attached picture in the question for mathematical formulation of the problem. :)

@ Henk Smid: Thanks for the attention. :)

Dear Hosein,

The book of Szilard has many solution to vibration of plates (http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471429899.html). Into this book, you find a closed solution for natural frequencies and mode shapes with expansion of sin for simple supported plates. This approaches use Ritz method. Can be useful for you. 

But  for your clamped plate, its better appeal to some article, for example:

Vijayakumar, K., e G.K. Ramaiah. 1978. “Analysis of vibration of clamped square plates by the Rayleigh-Ritz method with asymptotic solutions from a modified Bolotin method.” Journal of Sound and Vibration 56 (1): 127–35. doi:10.1016/0022-460X(78)90575-8.

Bhat, R.B. 1985. “Natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh-ritz method.” Journal of Sound and Vibration 102 (4): 493–99. doi:10.1016/S0022-460X(85)80109-7.

Dear Marcus

Thanks for the references. I am just looking for exact closed form solution not an approximate one. I knew that this problem has many approximate closed form solutions by Ritz and Galerkin methods.