Consider the inhomogeneous equation

\[\Delta^2 u = f(\mathbf{x}),\qquad \mathbf{x}\in\Omega \]

for some domain \(\Omega\subset\mathbf{R}^d \). Here \(\Delta\) is the Laplacian. The boundary conditions to be employed are

\[u(\mathbf{x})=\Delta u(\mathbf{x})=0, \quad \mathbf{x}\in\partial\Omega \]

The important case for us is \(d=2\), i.e. \(\mathbf{x}=(x,y) \). Begin by letting \(\Omega\) be a rectangle. In Cartesian coordinates we have

\[\Delta^2 u = u_{xxxx}+2u_{xxyy}+u_{yyyy} \]

Other possibilities include \(\Omega\) being a disc with radius \(1\), which is best solved in polar coordinates.

Google Biharmonic equation, thin plate, biharmonic operator, Laplacian.

Challenges High order derivatives.