One formulation of the Calabi conjecture, proved by Yau, is the following (p.100 in Joyce: Compact Manifolds with Special Holonomy):
Let $M$ be a compact, complex manifold of complex dimension $m$, $g$ a Kähler metric on $M$. Let $f$ be a smooth real function on $M$, and $A>0$ defined by $A\int_Me^f d vol_g=vol_g(M)$. Then there exists a unique real function $\phi$ such that
(i) $\int_M \phi d vol_g=0$
(ii) $(\omega+dd^c \phi)^m=Ae^f \omega^m$
I would like to compute an approximate solution $\phi_h$ of $\phi$ using some finite element method, and then have an estimate
$$|| \nabla \nabla \nabla (\phi_h-\phi)||_{L^\infty} \leq F(f,\phi_h,h),$$ where $F$ is some function depending on $M$ and my chosen discretisation of $M$.
For Laplace's equation, we have en estimate of the form $|| \nabla (\phi_h-\phi)||_{L^2} \leq F(f,\phi_h)$ as equation 2.6.10 in Repin: A posteriori estimates for partial differential equations. This is different from the estimate I want in two ways: (1) it is a different PDE, and (2) it only has one $\nabla$ and not three.
To address difference 1:
Question: Is there a blueprint for proving a posteriori estimates (in whatever norm) for a specific PDE? Where can I learn about this?
To address difference 2, I may be able to bump up an $L^2$-estimate to a $C^3$-estimate using elliptic regularity. I'm not sure about this, but this would be a different question.