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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {W^{1,p}} of planar diffeomorphisms. We introduce a concept called 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>W</m:mi> <m:mi>id</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>δ</m:mi> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {g_{\delta}:[-1,1]^{2}\to\mathbb{R}^{2}} and very small open sets <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>U</m:mi> <m:mi>δ</m:mi> </m:msub> </m:math> {U_{\delta}} such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>δ</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>u</m:mi> </m:mrow> </m:math> {g_{\delta}=u} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>∖</m:mo> <m:msub> <m:mi>U</m:mi> <m:mi>δ</m:mi> </m:msub> </m:mrow> </m:math> {[-1,1]^{2}\setminus U_{\delta}} . Also, we introduce the so-called the three-curve condition, which is an adaption of the NCL condition of [D. Campbell, A. Pratelli and E. Radici, Comparison between the non-crossing and the non-crossing on lines properties, Journal of Mathematical Analysis and Applications 498 2021, Article ID 124956] for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {u^{-1}} instead of for u , and prove that a map is the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">