h-principle for the Monge-Ampère equations

We discuss $h$-pricniple for the solutions to the Monge-Ampère equation in two dimensions. Namely, for a simply connected domain $\Omega\subset \mathbb R^2$, and any data $f: \Omega \to \mathbb R$, we show that the very weak $C^{1,\alpha}$ solutions to the equation

$$ \det D^2 u = f \quad \mbox{in} \,\, \Omega $$

are dense in the set of all continuous functions below the regularity threshold <1/7. We will also prove that the statement fails in the regularity regime $\alpha >2/3$ for the same class of very weak solutions.

This h-principle statement is a consequence of the convex integration methods and parallels similar results such as the Nash-Kuiper $C^1$ isometric embedding theorem and existence of continuous solutions with anomalous dissipation to Euler equations (due to De Lellis and Székelyhidi).