Finding a minimization problem corresponding to a PDE

Question

I was trying to find an equivalent minimization problem to the following PDE in $\Omega \subset \mathbb{R}^2$

$$ \Delta^2 u-\nabla \cdot (k(x,y) \nabla u)+\lambda u = f(x,y) $$ where $\lambda >0 $ and $k(x,y) >0$.

With the following boundary conditions:

$$\frac{\partial u}{\partial n} =0 \text{ and } \frac{\partial }{\partial n} (\Delta u) - k(x,y) \frac{\partial u}{\partial n} =u\text{ on }\partial \Omega$$

I was also wondering whether the number of BC's are enough for a unique solution, as we have a fourth order PDE.

Thanks in advance for your helpful hints.

Answer 1

Assuming that there exist a minimization problem for the given PDE. Then it can be found by computing: $$J(u) = \int_\Omega uLu-uf~d\Omega$$ where $L$ is the operator of the PDE. $$Lu = \Delta^2u-\nabla\cdot(k\nabla u) + \lambda u$$ You can use your boundary conditions and partial integration + Gauß to formulate the problem in the classical way with the norms and so on. If you need more steps let me know.

Answer 2

If there exists a solution it will make the following $0$: $$u_o=\min_{u}\left\{\|\Delta^2u-\nabla \cdot (k(x,y)\nabla u) + \lambda u - f(x,y)\|_2^2 +\\+ \left\|\frac{\partial u}{\partial n}\right\|_2^2 +\left\|\frac{\partial}{\partial n}(\Delta u) - k(x,y)\frac{\partial u}{\partial n} - u\right\|_2^2\right\}$$