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

Abstract

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 $g_δ:[-1,1]^2\to \mathbb{R}^2$ and very small open sets $U_δ$ such that $g_δ = u$ on $[-1,1]^2 \setminus U_δ$. 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}.

Figures (12)

• Figure 1: If a mapping is not injective then there may be many ways to approximate its restriction to a curve by injective maps which may have very different geometry. Here we see an 'S' shape in blue and a 'Z' shape in red, both of which could be the image of an injective approximation of a map that maps a segment onto another 3 times.
• Figure 2: Let, for example $C_1$ be the blue set on the right and $C_2$ be the green set on the right. Let $\gamma$ be the black curve in the shaded area by the boundary where $u$ is identity. If there is some point $(x,y)\in u^{-1}(C_1) \cap u^{-1}(C_2)$ then necessarily $u^{-1}(C_1\cup C_2 \cup \gamma)$ must disconnect the plane (shaded red component).
• Figure 3: We construct our grid $\Gamma_1$ by splitting the square by the black lines. The grid $\Gamma_2$ is made by including the blue lines. The grid $\Gamma_3$ is made by adding red lines. We make sure that neighboring lines in $\Gamma_k$ have distance close to $2^{-k}$ by adding the correct number of lines between neighbors, either 0, 1 or 2 lines are added.
• Figure 4: The definition of $\tilde{\varphi}_{\delta}$ on $V$ using $\varphi_{\delta}$.
• Figure 5: Given rectangles of $\Gamma_k$, we make $\Gamma_{k+1}$ by dividing them into smaller rectangles each of which have side length approximately $2^{-k-1}$. In some cases we split a side in three parts, in some cases we split it in two and if it is short, we do not split it.

Theorems & Definitions (67)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem A
• Remark 1.4
• proof : Proof that $u$ satisfies WM $\Rightarrow$ $u\in \overline{H\cap W^{1,p}_{\operatorname{id}}((-1,1)^2, \mathbb R^2)}$
• Theorem B
• proof
• Definition 2.1
• Proposition 2.2: The existence of many suitable grid systems