I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline understanding and solution
The problem is essentially an interpolation problem with fixed boundary conditions, specifically I want to use thin plate spline.
$$ \left\{ \begin{array}{l} \min_{u} \int_\Omega \sum \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2 dx dy \\ f(x_i,y_i) = z_i && 1 \leq i \leq N\\ f(x,y) = g(x,y) && (x,y) \in \partial \Omega \end{array} \right. $$
The problem above without boundary conditions I think I've attempted to solve it in the linked question. The use of boundary condition I am not sure how to handle.
Also $\Omega$ is an open connected set in $\mathbb{R}^2$ also $f : \Omega \to \mathbb{R}$ is a twice differentiable function at least.
In my previous question I've made the assumption of working in a Reproducing Kernel Hilbert Space. However I am not sure on how to model this constraint so I can use it in the variational problem and later maybe use the Lagrange multiplier method.
