Biharmonic equation boundary conditions

Question

Suppose I have a region $\Omega$ in the plane and I want to solve the biharmonic equation $$\Delta^2 f = 0$$ over $\Omega$. I must specify two boundary conditions. The simplest would be if I prescribed $f = f_0$ and $\Delta f = g_0$ on $\partial \Omega$: then I can decompose the biharmonic equation into two Poisson equations

\begin{align*} \Delta f &= h \\ \Delta h &= 0 \end{align*} with Dirichlet boundary conditions.

However, in practice, you usually see "Dirichlet" boundary conditions $$f = f_0, \qquad \nabla f \cdot \hat{n} = g_0.$$ Is there any way to break the biharmonic equation into a pair of Poisson equations, for these boundary conditions? You can still introduce an auxiliary function $h$, but would need some way of transforming the normal derivative boundary condition on $f$ into a condition on $h$...

Answer 1

The answer is negative. This is not something I can prove, but if such decoupling was available, it would be all over the numerical PDE literature. Instead, I see:

Due to the mixed boundary conditions the system of equations in Eqs. (15) cannot be decoupled.

in A highly accurate numerical solution of a biharmonic equation by M. Arad, A. Yakhot, G. Ben-Dor.

Guo Chen, Zhilin Li, and Ping Lin in A fast finite difference method for biharmonic equations on irregular domains... also comment on the difficulty of dealing with boundary conditions in your post. They develop the following method:

1. Begin with a guess for boundary values of $\Delta f$.
2. Use Poisson solvers to first compute $\Delta f$, and then $f$.
3. Compare $\partial f/\partial n$ with the desired boundary values.
4. Based on 3, adjust the guess in step 1, and repeat.