Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity

We present three novel classifications of the weak sequential (and strong) limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called the QM condition which is a kind of separation property for pre-images of closed connected sets and show that $u$ satisfies this property exactly when it is the limit of Sobolev homeomorphisms. Further, we prove that $u\in W^{1,p}_{\operatorname{id}}((-1,1)^2,\mathbb{R}^2)$ is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings $g_{\delta}:[-1,1]^2\to \mathbb{R}^2$ and very small open sets $U_{\delta}$ such that $g_{\delta} = u$ on $[-1,1]^2 \setminus U_{\delta}$. Also, we introduce the so-called three curve condition, which is in some sense reminiscent of the NCL condition of \cite{CPR} but for $u^{-1}$ instead of for $u$, and prove that a map is the $W^{1,p}$ limit of planar Sobolev homeomorphisms exactly when it satisfies this property. This improves on results in \cite{DPP} answering the question from \cite{IO2}.