# The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions

I am interested in solving the following biharmonic eigenvalue problem.

$$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ & x = a & \Psi = 0 & \dfrac{\partial \Psi }{\partial x} = 0 \\ & x = - a & \Psi = 0 & \dfrac{\partial \Psi }{\partial x} = 0 \\ & y = b & \Psi = 0 & \dfrac{\partial \Psi }{ \partial y} = 0 \\ & y = - b & \Psi = 0 & \dfrac{\partial \Psi }{ \partial y} = 0 \end{array}$$

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.

2) Assuming the existence, How can one compute these eigenvalues and eigenfunctions?

1) This question received more attention on Mathematics Stack Exchange. You can take a look over there too.

2) A proof for the existence is given there by TKS.

• Why are you interested in this problem? Oct 7 '15 at 14:41
• It's the vibrating modes of a clamped (square) plate, much harder than those of a plate simply supported on its sides. There seems to be no known explicit values of the frequencies, and if the eigenfunctions were separable I'm quite sure it would be known. Not very encouraging, I know... Oct 7 '15 at 15:07
• Have you read the following paper and references therein ? rspa.royalsocietypublishing.org/content/462/2068/1107 Oct 7 '15 at 16:16
• I believe Meleshko's work (either 1998 or earlier) refered in above paper includes extensive discussion of using eigenfunctions of biharmonic operator to solve the PDE. Oct 7 '15 at 16:21
• @H.R. Not a bad question, but probably a hard one! (I once computed the eigenvalues of the 1-dim version, solutions of $\tan\lambda=\tanh\lambda$ if I remember well. But for the clamped plate the belief of engineers is worth a thought, I think.) Oct 8 '15 at 9:14

At least the existence of eigenvalues and eigenfunctions (question 1) is routine common knowledge. The first one is the minimizer of $\int (\Delta u)^2$ subject to $u\in H^2_0$ and $\int u^2\le1$ (which exists, due to the compacness of $H^2_0\hookrightarrow L^2$). What is hard is to compute/characterize them (question 2). Depending on what your exact needs are, numerical methods are worth considering (finite elements, Galerkin,...)