I am trying to derive the "[thin plate energy functional](https://en.wikipedia.org/wiki/Thin_plate_energy_functional)". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional


$$\iint\_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\right)^2+2\left(\frac{\partial^2 z}{\partial x\partial y}\right)^2+\left(\frac{\partial^2 z}{\partial y^2}\right)^2\right]\mathrm{d}x\,\mathrm{d}y\quad ?$$



The derivation of thin plate spline interpolation energy function？
It very much depends on what you consider a "derivation" and what you consider an "easy" one. You can start *assuming* that the energy density depends on $(\kappa\_1^2+\kappa\_2^2)g^{\frac 12}$, where $\kappa\_i$ are the principal curvatures, which can be considered quite natural from a physical point of view (in one dimension this corresponds to assuming that the energy density of a 1D elastic continuum depends quadratically on the curvature, and it leads to the non-linear theory of the Euler *Elastica*). Then you can deduce your integral functional as a linearized form of the problem, as is done in most textbooks (and also sketched in the wiki page you linked).


If instead you want more something like a "microscopic" derivation, then you'll need more sophisticated tools to make sense of the limit functional of a suitable sequence of 3D Cauchy-continuum energy functionals, when one of the dimensions goes to zero (and the coefficients are rescaled accordingly). A standard reference (in my view a very good one, as it covers the mathematical aspects in full rigor) is: Ciarlet, P. G. (1997). Mathematical Elasticity: Volume II: Theory of Plates. Elsevier.


Hope this helps.


Btw: I would not say "Given a thin plate $z=z(x,y)$", as $z$ is in fact the placement function.


